Namespace AutomorphicForm 2,336 theorems
Landmarks here: From cuspidal adelic eigensystems to classical weight-one cusp forms
— 1902 · AdelicTracePushforward 3 · ArchOccursInClassOf 1 · ArchWeightOne 2 · ClassSumGrowth 2 · ComplexIwasawa 12 · CuspidalConstituent 103 · CuspidalSpectrum 65 · CyclicBaseChangeLifting 2 · GL2Real 22 · GL2Twisted 12 · HeckeEigensystem 7 · IdeleChar 1 · IsArchTestFactor 2 · IsCuspidalFn 3 · IsFactorizableTestFn 3 · IsFinTestFactor 1 · IsGL2RealKTypeModule 1 · IsInducedSection 2 · IsIsotypicCuspFormAt 2 · IsKfSmooth 3 · IsOrbitalIntegralOn 2 · IsRegularSemisimple 1 · IsSlabProfile 2 · IsTwistedOrbitalIntegralOn 2 · IsTwistedWeightedOrbitalIntegralOn 1 · IsUnitFactorizableAbove 1 · IsWeightedOrbitalIntegralOn 1 · LocalFunctionSpace 8 · LocalIntertwining 19 · LocalWeightedOrbital 6 · PseudoEisensteinSlab 1 · RankinSelberg 18 · RealIwasawa 7 · SatakeCombination 7 · SiegelCovering 3 · SmoothCusp 1 · SmoothCuspRealizationAt 26 · SplitPlace 4 · StandardKernel 1 · TransversalMeasure 6 · TwistedBruhat 30 · WeylIntegrable 6 · WhittakerModel 21 · WindingDatum 5 · WindowedSiegel 6
directly in AutomorphicForm 1902
- landmark From cuspidal adelic eigensystems to classical weight-one cusp forms
AutomorphicForm.exists_weightOne_cuspForm_of_isCusp_viaCompactCuspNotion6 below · cited by 1 · depth 10 - Finite set of ideles controlling bottom-row content classes
AutomorphicForm.exists_finset_forall_exists_mem_valued_eq_max_and_contentHomFin_mul_sq_eq1 below · cited by 1 · depth 12 - Class criterion for an upper-triangular adelic decomposition
AutomorphicForm.exists_upperTriangular_globalPoints_mul_mul_scalar_mul_finIdeleDiag_inv_mem_finiteIntegralGL20 below · cited by 1 · depth 12 - Weight-one cusp form from a cuspidal Hecke eigensystem over ℚ
AutomorphicForm.exists_weightOne_cuspForm_of_isCusp_viaGeneralCuspNotion6 below · cited by 1 · depth 13 - Boundedness upgrade for cusp realizations on covering Siegel windows
AutomorphicForm.isArithBoundedGenuineCuspRealizable_of_isArithGenuineCuspRealizable_of_coversModCentre76 below · cited by 3 · depth 13 - Right convolution by a factorizable test function: continuity and Cᵈ⁺¹ regularity
AutomorphicForm.continuous_rightConv_and_contDiff_of_isFactorizableTestFn2 below · cited by 61 · depth 14 - Galois conjugation of a cusp-realizable Hecke eigensystem
AutomorphicForm.exists_isArithBoundedGenuineCuspRealizable_eq_comap_galRestrict1 below · cited by 1 · depth 14 - Bounded genuine realizability of a degree 2 or 3 base-change descent
AutomorphicForm.exists_isArithBoundedGenuineCuspRealizable_formalBaseChange_of_isConstantOnFibers_of_finrank_two_or_three_of_coversModCentre3,124 below · cited by 2 · depth 14 - Non-vanishing right convolution with a level-N factorisable test function
AutomorphicForm.exists_isFactorizableTestFn_rightConv_ne_zero_of_levelOne_invariant1 below · cited by 10 · depth 14 - Right convolution of a cuspidal function is Siegel-window bounded
AutomorphicForm.isBoundedOnSiegelWindows_rightConv_of_isCuspAutomorphicFnAt_of_coversModCentre70 below · cited by 12 · depth 14 - Right convolution preserves cuspidality, smoothness, level and Hecke eigenvalues
AutomorphicForm.isCuspidalFn_isKfSmooth_levelInvariant_isHeckeCosetEigenfunctionAt_rightConv_of_isFactorizableTestFn_of_support_subset2 below · cited by 11 · depth 14 - Lucas recursion computes power sums αⁿ+βⁿ
AutomorphicForm.satakePow_add_pow0 below · cited by 3 · depth 14 - Integrability and summability of adelic GL₂ Whittaker coefficients
AutomorphicForm.whittakerCoefficientIntegrable_and_summable_of_isKfSmooth_of_contDiff_mixedSpace12 below · cited by 14 · depth 14 - Factorisable test functions are continuous and compactly supported
AutomorphicForm.continuous_and_hasCompactSupport_of_isFactorizableTestFn0 below · cited by 170 · depth 15 - Continuity of the unipotent embedding into GL₂(R)
AutomorphicForm.continuous_unipotentGL20 below · cited by 122 · depth 15 - Raising the lower determinant bound of a centre-cut Siegel window
AutomorphicForm.coversModCentre_and_isArithGenuineCuspRealizable_of_le_of_lt_of_coversModCentre0 below · cited by 6 · depth 15 - Base change of an Eisenstein Hecke table is Eisenstein
AutomorphicForm.exists_agreesAwayFromFinite_formalBaseChange_eisensteinTableOf7 below · cited by 1 · depth 15 - Descent of a fibre-constant eigensystem to a formal base change
AutomorphicForm.exists_formalBaseChange_of_isConstantOnFibers_of_finrank_two_or_three_of_coversModCentre3,122 below · cited by 1 · depth 15 - Weight-one realization of Theta from a translate-span witness
AutomorphicForm.exists_isGenuineCuspRealizationAt_archWeightOne_isArchHolomorphicAt_iff_of_isInTranslateSpanOn_of_finite32 below · cited by 1 · depth 15 - Windowed adelic realization of a weight-one primitive form
AutomorphicForm.exists_isGenuineCuspRealizationAt_hasNewvectorConductor_adelicSpan_factorization_of_isPrimitiveForm_weightOne52 below · cited by 1 · depth 15 - Global additive characters of A_F are dilates of ψ_F
AutomorphicForm.exists_ne_zero_forall_eq_stdAddChar_mul_of_isGlobalAddChar10 below · cited by 13 · depth 15 - High-height boundedness of a smoothed cuspidal function
AutomorphicForm.exists_norm_rightConv_mul_le_of_lt_localHeight_of_isCuspAutomorphicFnAt_of_coversModCentre67 below · cited by 1 · depth 15 - Strong multiplicity one: mean-square approximation by right translates
AutomorphicForm.exists_setLIntegral_sub_sum_translate_sq_lt_of_agreesAwayFromFinite_of_coversModCentre503 below · cited by 2 · depth 15 - Rankin's logarithmic second-moment bound for Hecke eigenvalues
AutomorphicForm.exists_tsum_norm_a_sq_mul_rpow_absNorm_le_log_of_isArithGenuineCuspRealizable715 below · cited by 1 · depth 15 - Siegel finiteness for ample centre-cut Siegel sets
AutomorphicForm.finite_setOf_exists_globalPoints_mul_mem_image_centreCutSiegelSetAmple4 below · cited by 2 · depth 15 - Transitivity of formal base change of Hecke eigensystems
AutomorphicForm.formalBaseChange_formalBaseChange0 below · cited by 2 · depth 15 - Transporting arithmetic bounded genuine cusp-realizability to Siegel windows
AutomorphicForm.isArithBoundedGenuineCuspRealizable_of_isArithBoundedGenuineCuspRealizable_of_pos_of_pos0 below · cited by 1 · depth 15 - Twisting a realizable eigensystem by a power of the norm
AutomorphicForm.isArithGenuineCuspRealizable_twist_rpow_absNorm10 below · cited by 2 · depth 15 - A genuine cusp realization excludes Eisenstein Hecke tables
AutomorphicForm.not_agreesAwayFromFinite_eisensteinTableOf_of_isArithGenuineCuspRealizable_of_coversModCentre211 below · cited by 2 · depth 15 - No genuine cusp realizations over a Siegel window with non-positive height floor
AutomorphicForm.not_isArithGenuineCuspRealizable_of_nonpos_of_lt_of_coversModCentre0 below · cited by 16 · depth 15 - Archimedean K-types of a class: parity or discrete series
AutomorphicForm.archOccursInClassOf_archWeightChar_iff_parity_or_discreteSeries_of_coversModCentre403 below · cited by 1 · depth 16 - Left B(K)-invariance of the box constant term
AutomorphicForm.constantTerm_adelicBox_globalPoints_mul_of_mem_borelSubgroup4 below · cited by 15 · depth 16 - Unipotent invariance of the box constant term on GL₂(A_K)
AutomorphicForm.constantTerm_adelicBox_unipotentGL2_mul0 below · cited by 14 · depth 16 - Passage from ample to plain centre-cut Siegel windows
AutomorphicForm.exists_centreCutSiegelSetAmple_coversModCentre_and_realizations_and_approximation_of_coversModCentre104 below · cited by 1 · depth 16 - Entire continuation of the twisted partial L-function
AutomorphicForm.exists_differentiable_hasProd_eulerProduct_twist_of_isArithGenuineCuspRealizable189 below · cited by 3 · depth 16 - Partial Rankin–Selberg L-function: simple pole at s=1
AutomorphicForm.exists_finset_forall_lt_one_meromorphicOn_meromorphicOrderAt_one_eq_neg_one_analyticAt_hasProd_rsEulerPoly_self702 below · cited by 1 · depth 16 - Hecke eigenvalues persist under translation to another level
AutomorphicForm.exists_forall_isHeckeCosetEigenfunctionAt_finTranslateSum_of_levelOne_invariant1 below · cited by 1 · depth 16 - Archimedean-finite smoothing inside an isotypic cusp space
AutomorphicForm.exists_isArchKFinite_tendsto_and_setLIntegral_le_of_mem_isotypicCuspSubmodule18 below · cited by 1 · depth 16 - A fundamental domain inside a determinant norm slab for GL₂(F)
AutomorphicForm.exists_isFundamentalDomain_globalPoints_range_restrict_ideleNorm_det_Icc7 below · cited by 33 · depth 16 - Nonzero level combinations of translates, and their mean-square density
AutomorphicForm.exists_levelInvariant_finTranslateSum_ne_zero_and_dense_of_isInTranslateSpanOn_of_finite20 below · cited by 1 · depth 16 - Invariant mean square comparable to mass over a Siegel window
AutomorphicForm.exists_measure_lintegral_translate_eq_mul_and_setLIntegral_le_mul_of_coversModCentre_of_finite15 below · cited by 2 · depth 16 - Cyclic base change descent in degree 2 or 3
AutomorphicForm.exists_mem_cuspClasses_of_twistedCutTrace_ne_zero_of_finrank_two_or_three3,105 below · cited by 1 · depth 16 - Strong multiplicity one on an ample Siegel window for GL₂
AutomorphicForm.exists_setLIntegral_sub_sum_translate_sq_lt_of_agreesAwayFromFinite_of_coversModCentre_ample133 below · cited by 2 · depth 16 - Transporting a cusp realization to Siegel-window production pins
AutomorphicForm.exists_smoothCuspRealizationAt_productionPinsOf_toFun_eq_of_isBoundedOnSiegelWindows_of_coversModCentre0 below · cited by 1 · depth 16 - Non-vanishing twisted cut trace for a fibre-constant eigensystem
AutomorphicForm.exists_twistedCutTrace_ne_zero_of_pos_of_isArithGenuineCuspRealizable_of_isConstantOnFibers361 below · cited by 1 · depth 16 - Siegel window mass bounded by fundamental domain mass
AutomorphicForm.exists_window_mass_le_mul_domain_mass_of_isArchKFinite_of_mem_isotypicCuspSubmodule490 below · cited by 1 · depth 16 - Whittaker–Fourier expansion of a continuous unipotent slice
AutomorphicForm.hasSum_whittakerCoefficient5 below · cited by 15 · depth 16 - Principal ideles have idele norm one, via det on GL₂
AutomorphicForm.ideleNorm_det_globalPoints3 below · cited by 170 · depth 16 - Weight-one holomorphy from mean-square approximation by holomorphic translates
AutomorphicForm.isArchHolomorphicAt_of_forall_exists_setLIntegral_sub_sum_holomorphic_translate_sq_lt6 below · cited by 2 · depth 16 - Factorizable test functions are stable under left translation
AutomorphicForm.isFactorizableTestFn_comp_inv_mul_of_isFactorizableTestFn0 below · cited by 18 · depth 16 - Isotypic cusp space on a covering window embeds into the slab domain space
AutomorphicForm.isotypicCuspSubmodule_le_of_coversModCentre_of_isFundamentalDomain_slab12 below · cited by 8 · depth 16 - Extending the determinant window of an L² automorphic function
AutomorphicForm.memLp_iUnion_centreCutSiegelSet_of_detWindow_le0 below · cited by 7 · depth 16 - Square-integrability forces contracting archimedean central character
AutomorphicForm.norm_apply_archCentralUnit_lt_one_of_memLp_of_coversModCentre2 below · cited by 8 · depth 16 - Cuspidality bound for int φ(y)f(x⁻¹y) on GL₂(A_K)
AutomorphicForm.norm_integral_mul_le_mul_setIntegral_norm_of_isCuspidalFn2 below · cited by 4 · depth 16 - Decay of rational unipotent sums minus box average in Siegel sets
AutomorphicForm.norm_tsum_sub_average_le_mul_inv_archHeight_pow_of_isFactorizableTestFn57 below · cited by 5 · depth 16 - Right translation of a right convolution on GL₂(A_K)
AutomorphicForm.rightConv_apply_mul_eq_rightConv_comp_inv_mul_apply0 below · cited by 48 · depth 16 - Square-mass bound on unipotent sweeps high in a Siegel set
AutomorphicForm.setLIntegral_nnnorm_sq_le_mul_archHeight_pow_mul_setLIntegral_of_isLsXiFunction_of_coversModCentre5 below · cited by 1 · depth 16 - Whittaker coefficients: W_α(g)=W₁(diag(α,1)g)
AutomorphicForm.whittakerCoefficient_eq_whittakerCoefficient_one_globalPoints_diagOne_mul3 below · cited by 12 · depth 16 - Unipotent covariance of adelic Whittaker coefficients on GL₂
AutomorphicForm.whittakerCoefficient_unipotentGL2_mul0 below · cited by 54 · depth 16 - Finiteness of Haar measure of a slab fundamental domain
AutomorphicForm.adelicGLHaar_inter_setOf_ideleNorm_det_mem_Icc_lt_top_of_isFundamentalDomain15 below · cited by 81 · depth 17 - Casimir dictionary for archimedean occurrence at a real place
AutomorphicForm.archOccursInClassOf_iff_archCasimirAt_of_coversModCentre526 below · cited by 1 · depth 17 - Covering mod centre is insensitive to the determinant window
AutomorphicForm.coversModCentre_iUnion_centreCutSiegelSet_of_detWindow0 below · cited by 3 · depth 17 - Uniqueness of a cuspidal constituent meeting given Hecke data
AutomorphicForm.eq_of_isCuspConstituent_of_cuspConstituentMeets_of_coversModCentre240 below · cited by 10 · depth 17 - Nonzero isotypic cusp spaces contain vectors of finite archimedean type
AutomorphicForm.exists_archTypeFamily_isotypicCuspSubmodule_inf_archCutSubmodule_ne_bot73 below · cited by 3 · depth 17 - Ample centre-cut Siegel windows still cover modulo centre
AutomorphicForm.exists_coversModCentre_centreCutSiegelSetAmple0 below · cited by 3 · depth 17 - Level-one cuspidal descent from a non-vanishing twisted cut trace
AutomorphicForm.exists_cuspClass_of_twistedCutTrace_ne_zero_of_areMatchingAt_symm_principalLevel_finrank_two_or_three3,064 below · cited by 1 · depth 17 - Entirety of the global Whittaker zeta integral on GL₂
AutomorphicForm.exists_differentiable_forall_integral_zetaIntegrand_whittakerCoefficient_unipotentAverage_eq141 below · cited by 1 · depth 17 - Archimedean characters at a real place are integral weight characters
AutomorphicForm.exists_eq_archWeightCharReal_of_hasArchCharacterAt_of_continuous0 below · cited by 2 · depth 17 - Finite central covering of determinant-norm slabs
AutomorphicForm.exists_finset_central_slab_covering_of_coversModCentre_centreCutSiegelSetAmple6 below · cited by 4 · depth 17 - Convolution operators realise endomorphisms of window isotypic cusp spaces
AutomorphicForm.exists_finset_convOp_eq_of_le_isotypicCuspSubmodule_inf_archCutSubmodule_of_coversModCentre347 below · cited by 1 · depth 17 - Finite central set suffices on a determinant slab
AutomorphicForm.exists_finset_globalPoints_mul_mul_centralScalar_mem_of_coversModCentre_of_ideleNorm_det_mem_Icc7 below · cited by 5 · depth 17 - Simple pole at s=1 of a partial Rankin–Selberg Euler product
AutomorphicForm.exists_finset_lt_one_meromorphicOn_meromorphicOrderAt_one_eq_neg_one_analyticAt_hasProd_rsEulerPoly_self701 below · cited by 1 · depth 17 - Determinant slabs are covered without the centre
AutomorphicForm.exists_finset_slab_covering_of_coversModCentre6 below · cited by 4 · depth 17 - Uniformly bounded multiplicity of finitely many ample Siegel translates
AutomorphicForm.exists_forall_ncard_setOf_globalPoints_mul_mem_iUnion_centreCutSiegelSetAmple_le5 below · cited by 6 · depth 17 - Decay bound for cuspidal functions on a centre-cut Siegel window
AutomorphicForm.exists_forall_norm_le_mul_prod_rpow_neg_of_hasDerivAt_chains_of_constantTerm_eq_zero_of_mem_idealBall32 below · cited by 1 · depth 17 - Uniform bound for right convolution on centre-cut Siegel windows
AutomorphicForm.exists_forall_norm_rightConv_le_mul_eLpNorm_of_isLsXiFunction_of_isCuspidalFn_of_isFundamentalDomain72 below · cited by 21 · depth 17 - Uniform L²-mass bound over a compact set
AutomorphicForm.exists_forall_setLIntegral_nnnorm_sq_le_mul_setLIntegral_of_isLsXiFunction_of_isCompact_of_coversModCentre2 below · cited by 4 · depth 17 - Uniform L² bound over compacta by fundamental-domain mass
AutomorphicForm.exists_forall_setLIntegral_nnnorm_sq_le_mul_setLIntegral_of_isLsXiFunction_of_isCompact_of_isFundamentalDomain9 below · cited by 4 · depth 17 - Twisting a cusp-realizable GL₂ eigensystem by a ray class character
AutomorphicForm.exists_isArithGenuineCuspRealizable_rayClassChar_twist_of_coversModCentre96 below · cited by 1 · depth 17 - A σ-stable enlargement of an archimedean type family
AutomorphicForm.exists_isContainedIn_forall_sigmaSectionActOn_mem_archCutSubmodule8 below · cited by 1 · depth 17 - Archimedean K-type profile of a cuspidal class at a real place
AutomorphicForm.exists_isGL2RealKTypeModule_archOccursInClassOf_iff_of_coversModCentre400 below · cited by 1 · depth 17 - K-finite approximate identity on the archimedean maximal compact subgroup
AutomorphicForm.exists_kernel_concentrating_translatesSpanFinite_maximalCompactAt0 below · cited by 1 · depth 17 - Local components of a global additive character of A_F
AutomorphicForm.exists_localComponents_of_isGlobalAddChar21 below · cited by 26 · depth 17 - Partial Rankin–Selberg product: meromorphy and pole rigidity
AutomorphicForm.exists_lt_one_meromorphicOn_hasProd_rsEulerPoly_and_agreesAwayFromFinite_of_meromorphicOrderAt_one_neg735 below · cited by 1 · depth 17 - Pole at s=1 of partial Rankin–Selberg Euler products
AutomorphicForm.exists_lt_one_meromorphicOn_hasProd_rsEulerPoly_self_and_meromorphicOrderAt_one_neg703 below · cited by 1 · depth 17 - The subgroups K^S shrink to 1 in GL₂(mathbb A_F)
AutomorphicForm.exists_maximalCompactAway_subset_of_mem_nhds_one0 below · cited by 4 · depth 17 - Measurable fundamental domain inside finitely many Siegel translates
AutomorphicForm.exists_measurableSet_isFundamentalDomain_subset_iUnion_integralWindowedSiegelSet_of_coversModCentre12 below · cited by 3 · depth 17 - Adelic Iwasawa decomposition for GL₂ over a number field
AutomorphicForm.exists_mem_adelicBorel_mul_eq1 below · cited by 63 · depth 17 - Adelic decomposition GL₂(A_ℚ)=GL₂(ℚ)· h· U₁(N)
AutomorphicForm.exists_mem_productionPinsCompact_U_mul_eq_rat5 below · cited by 7 · depth 17 - Moderate growth in det of a smoothed adelic cusp form
AutomorphicForm.exists_norm_rightConv_le_mul_max_ideleNorm_det_pow81 below · cited by 7 · depth 17 - Matching test functions for cyclic base change in degree 2 or 3
AutomorphicForm.exists_principalLevel_areMatchingAt_of_isUnitFactorizableAboveOfType_of_finrank_two_or_three77 below · cited by 1 · depth 17 - Weighted Petersson pairing: covariance, non-vanishing, sesquilinear form
AutomorphicForm.exists_sesqForm_eq_peterssonIntegral_of_isGenuineCuspRealizationAt_of_isFundamentalDomain12 below · cited by 11 · depth 17 - Smoothing a cusp realization by convolution with a test function
AutomorphicForm.exists_smoothCuspRealizationAt_toFun_eq_rightConv_of_isArithGenuineCuspRealizable79 below · cited by 1 · depth 17 - An entire, non-vanishing S-part torus zeta integral
AutomorphicForm.exists_unipotentAverage_rightConv_sPart_zetaIntegrand_entire_ne_zero118 below · cited by 1 · depth 17 - Nonvanishing of the first Whittaker coefficient
AutomorphicForm.exists_whittakerCoefficient_one_ne_zero9 below · cited by 11 · depth 17 - Siegel window mass bounded by ample window mass
AutomorphicForm.exists_window_mass_le_mul_ample_window_mass_of_mem_isotypicCuspSubmodule489 below · cited by 1 · depth 17 - Finite-dimensionality of level and archimedean cuts of cuspidal constituents
AutomorphicForm.finiteDimensional_inf_levelInvariantSubmodule_inf_archCutSubmodule_of_isCuspConstituent160 below · cited by 17 · depth 17 - Finite-dimensionality of isotypic cusp spaces of fixed archimedean type
AutomorphicForm.finiteDimensional_isotypicCuspSubmodule_inf_archCutSubmodule334 below · cited by 11 · depth 17 - Determinant twists preserve archimedean type χ
AutomorphicForm.hasArchType0_fnTwist0 below · cited by 1 · depth 17 - Unfolding a cuspidal integral along rational unipotents with a weight
AutomorphicForm.integral_mul_eq_integral_mul_weight_mul_tsum_sub_average_of_isCuspidalFn1 below · cited by 1 · depth 17 - Holomorphy at a real place under L²-approximation by translates
AutomorphicForm.isArchHolomorphicAt_of_forall_exists_setLIntegral_sub_sum_translate_sq_lt7 below · cited by 1 · depth 17 - Smooth functions supported on invertible entry matrices give archimedean test factors
AutomorphicForm.isArchTestFactor_of_contDiff_of_hasCompactSupport_of_tsupport_subset_isUnit_det0 below · cited by 5 · depth 17 - Automorphy transports from a Siegel window to a fundamental domain
AutomorphicForm.isAutomorphicFnAt_of_isFundamentalDomain_of_isAutomorphicFnAt_of_coversModCentre11 below · cited by 13 · depth 17 - Central character of a non-zero continuous L_{s,ξ}-function
AutomorphicForm.isIdeleClassChar_and_continuous_of_isLsXiFunction_of_continuous0 below · cited by 22 · depth 17 - Nonzero members of the isotypic cusp span are cusp forms
AutomorphicForm.isIsotypicCuspFormAt_of_mem_isotypicCuspSubmodule1 below · cited by 27 · depth 17 - Isotypic cusp forms with archimedean cut lie in cuspidal constituents
AutomorphicForm.isotypicCuspSubmodule_inf_archCutSubmodule_le_iSup_isCuspConstituent330 below · cited by 12 · depth 17 - Compact averaging preserves isotypy, gives K-finiteness, decreases mass
AutomorphicForm.mem_isotypicCuspSubmodule_and_isArchKFinite_and_setLIntegral_le_of_integral_maximalCompactAtHaar_mul15 below · cited by 1 · depth 17 - Zero x-window centre-cut Siegel sets never cover modulo centre
AutomorphicForm.not_coversModCentre_iUnion_centreCutSiegelSet_of_eq_zero0 below · cited by 2 · depth 17 - Galois twist stabilises a finite-dimensional isotypic cusp space
AutomorphicForm.sigmaSectionActOn_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_finiteDimensional8 below · cited by 1 · depth 17 - Pointwise convergence of averages against concentrating kernels
AutomorphicForm.tendsto_integral_maximalCompactAtHaar_mul_of_concentrating0 below · cited by 1 · depth 17 - Twisted convolution preserves the isotypic type-cut cusp space
AutomorphicForm.twistedConvOp_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_isUnitFactorizableAboveOfType83 below · cited by 1 · depth 17 - Transfer of unramified Whittaker data to a Schwartz–Bruhat average
AutomorphicForm.whittakerCoefficient_unipotentAverage_unramified_package84 below · cited by 1 · depth 17 - Non-vanishing Whittaker coefficient forces ψ unramified outside S
AutomorphicForm.addChar_eq_one_on_integers_off_of_whittakerCoefficient_ne_zero1 below · cited by 1 · depth 18 - Casimir eigenvalue tfrac k2(1-tfrac k2) for lowest-weight vectors
AutomorphicForm.archCasimirAt_eq_smul_of_lower_eq_zero_of_hasArchCharacterAt3 below · cited by 5 · depth 18 - Norm twists preserve archimedean occurrence in a cuspidal class
AutomorphicForm.archOccursInClassOf_hasArchCharacterAtZero_archCasimirAt_iff_twist_rpow_absNorm11 below · cited by 1 · depth 18 - Lowering annihilates a witness with Casimir eigenvalue k/2(1-k/2)
AutomorphicForm.archOccursInClassOf_lower_eq_zero_of_archCasimirAt_eq_smul_of_coversModCentre24 below · cited by 9 · depth 18 - Continuity of the unipotent average of φ * f
AutomorphicForm.continuous_unipotentAverage_rightConv87 below · cited by 3 · depth 18 - Continuity of the adelic Whittaker coefficient in g
AutomorphicForm.continuous_whittakerCoefficient0 below · cited by 14 · depth 18 - Determinant-window transfer for centre-cut Siegel coverings
AutomorphicForm.coversModCentre_and_archOccursInClassOf_iff_of_detWindow_le9 below · cited by 6 · depth 18 - Raising the lower determinant bound preserves covering modulo the centre
AutomorphicForm.coversModCentre_of_le_of_lt_of_coversModCentre0 below · cited by 8 · depth 18 - Covering centre-cut Siegel window with c ≤ 0 forces vanishing
AutomorphicForm.eq_zero_of_isAutomorphicFnAt_of_continuous_of_nonpos_of_lt_of_coversModCentre0 below · cited by 7 · depth 18 - Determinant slabs are covered with finitely many central ideles
AutomorphicForm.exists_finset_central_slab_covering_of_coversModCentre6 below · cited by 17 · depth 18 - Right convolution realises every endomorphism of an isotypic cusp space
AutomorphicForm.exists_finset_convOp_eq_of_le_isotypicCuspSubmodule_inf_archCutSubmodule_of_isFundamentalDomain346 below · cited by 1 · depth 18 - Finitely many cuspidal constituents meet a fixed Hecke eigensystem
AutomorphicForm.exists_finset_isCuspConstituent_le_iSup_of_cuspConstituentMeets241 below · cited by 4 · depth 18 - Partial Rankin–Selberg product with a pole at s=1
AutomorphicForm.exists_finset_lt_one_meromorphicOn_analyticAt_hasProd_rsEulerPoly_self_and_eval_inv_absNorm_ne_zero702 below · cited by 2 · depth 18 - Partial Rankin–Selberg product for two cusp-realizable eigensystems
AutomorphicForm.exists_finset_lt_one_meromorphicOn_hasProd_rsEulerPoly_and_agreesAwayFromFinite_of_meromorphicOrderAt_one_neg734 below · cited by 1 · depth 18 - Casimir scalar at a real place: rigidity and regular witnesses
AutomorphicForm.exists_forall_archCasimirAt_eq_and_archOccursInClassOf_isArchSmoothAt_of_coversModCentre354 below · cited by 8 · depth 18 - A uniform central exponent at a real place
AutomorphicForm.exists_forall_archOccursInClassOf_and_centralExponent7 below · cited by 5 · depth 18 - Half-plane convergence of the GL(2) Whittaker zeta integral
AutomorphicForm.exists_forall_integrable_zetaIntegrand_whittakerCoefficient_unipotentAverage116 below · cited by 1 · depth 18 - Uniform convolution bound for smooth cusp forms on a Siegel window
AutomorphicForm.exists_forall_norm_rightConv_le_mul_eLpNorm_of_isSmoothCuspAutomorphicFnAt_of_coversModCentre68 below · cited by 10 · depth 18 - Smoothed cusp forms are bounded on determinant slabs
AutomorphicForm.exists_forall_norm_rightConv_le_of_ideleNorm_det_mem_Icc78 below · cited by 12 · depth 18 - Square-mass bound on Siegel sets against a slab fundamental domain
AutomorphicForm.exists_forall_setLIntegral_nnnorm_sq_le_mul_archHeight_pow_mul_setLIntegral_of_isLsXiFunction_of_isFundamentalDomain14 below · cited by 1 · depth 18 - Whittaker coefficients on a window vanish outside one fractional ideal
AutomorphicForm.exists_fractionalIdeal_forall_whittakerCoefficient_eq_zero_of_not_mem_of_forall_mul_idealBall_eq15 below · cited by 1 · depth 18 - Archimedean matching test factors for prime-degree base change
AutomorphicForm.exists_isArchTestFactor_isArchFactorBiFinite_areMatchingArch_of_algHom6 below · cited by 1 · depth 18 - Archimedean matching factor when L does not embed in K_∞
AutomorphicForm.exists_isArchTestFactor_isArchFactorBiFinite_areMatchingArch_of_isEmpty_algHom49 below · cited by 1 · depth 18 - Conjugation-invariant smooth bump with prescribed inversion symmetry
AutomorphicForm.exists_isArchTestFactor_nonneg_and_tsupport_subset_and_conj_invariant_and_flat4 below · cited by 7 · depth 18 - Cusp-realizable eigensystem realised in a single cuspidal constituent
AutomorphicForm.exists_isCuspConstituent_isIsotypicCuspFormAt_mem_archCutSubmodule_of_isArithGenuineCuspRealizable338 below · cited by 3 · depth 18 - Existence of a fundamental domain for GL₂(F)backslashGL₂(A_F)
AutomorphicForm.exists_isFundamentalDomain_globalPoints_range8 below · cited by 11 · depth 18 - Real-place types of a Hecke class form an irreducible K-type module
AutomorphicForm.exists_isGL2RealKTypeModule_archOccursInClassOf_iff_isArchLoweringAnnihilatedAt_of_coversModCentre399 below · cited by 1 · depth 18 - Local matching functions for prime-degree base change of GL₂
AutomorphicForm.exists_isLocalTestFn_areMatchingLocal_of_algHom4 below · cited by 1 · depth 18 - Local matching functions at a non-split place, degree 2 or 3
AutomorphicForm.exists_isLocalTestFn_areMatchingLocal_of_isEmpty_algHom23 below · cited by 1 · depth 18 - Approximate identity of prescribed level and archimedean types
AutomorphicForm.exists_isUnitFactorizableAboveOfType_tendsto_rightConv_of_mem_archCutSubmodule1 below · cited by 5 · depth 18 - Iwasawa formula for the T(K)N(A)-quotient measure
AutomorphicForm.exists_lintegral_rationalTorusUnipotentQuotientMeasure_eq_mul_setLIntegral_iwasawa18 below · cited by 12 · depth 18 - Principal congruence cuspidal classes descend to `levelOne` classes
AutomorphicForm.exists_mem_cuspClasses_levelOne_of_mem_cuspClasses_principalLevel97 below · cited by 3 · depth 18 - Cuspidal transfer of twisted cut trace, degree two or three
AutomorphicForm.exists_mem_cuspClasses_principalLevel_of_twistedCutTrace_ne_zero_of_areMatchingAt_symm3,063 below · cited by 1 · depth 18 - A finite-measure neighbourhood where the zeta integrand stays nonzero
AutomorphicForm.exists_nhd_whittakerCoefficient_diagOne_sPartMeasure_lt_top2 below · cited by 1 · depth 18 - Two-sided torus decay of a smoothed cuspidal unipotent average
AutomorphicForm.exists_norm_unipotentAverage_rightConv_diagOne_mul_le_min_ideleNorm_pow92 below · cited by 1 · depth 18 - Integration by parts bound for a Whittaker coefficient
AutomorphicForm.exists_norm_whittakerCoefficient_le_mul_of_hasDerivAt_unipotentGL2_of_forall_norm_le7 below · cited by 3 · depth 18 - Rapid decay of the first Whittaker coefficient of a smoothed cusp form
AutomorphicForm.exists_norm_whittakerCoefficient_rightConv_diagOne_mul_le_ideleNorm_rpow_neg_of_one_le92 below · cited by 2 · depth 18 - Left GL₂(ℚ)- and right K₁(N)-invariant functions agree
AutomorphicForm.ext_of_invariant_of_forall_glFin_eq_one_rat5 below · cited by 4 · depth 18 - Non-zero convolution eigenspaces of cusp forms are finite-dimensional
AutomorphicForm.finiteDimensional_of_forall_mem_rightConv_eq_smul76 below · cited by 2 · depth 18 - Torus Whittaker expansion of a smoothed adelic cusp form
AutomorphicForm.hasSum_whittakerCoefficient_one_diagOne_principalIdeles_unipotentAverage103 below · cited by 1 · depth 18 - Averaging over the maximal compact preserves the isotypic cusp space
AutomorphicForm.integral_maximalCompactAtHaar_mul_mem_isotypicCuspSubmodule13 below · cited by 1 · depth 18 - Central translation invariance of the unipotent-quotient integral
AutomorphicForm.integral_unipotentQuotient_out_mul_of_central3 below · cited by 1 · depth 18 - Archimedean K-finiteness of kernel averages over K_∞
AutomorphicForm.isArchKFinite_integral_maximalCompactAtHaar_mul_of_translatesSpanFinite0 below · cited by 1 · depth 18 - Lowest weight and y⁻¹-holomorphy via the lowering operator
AutomorphicForm.isArchLowestWeightAt_iff_and_isArchHolomorphicAt_iff_lower_eq_zero_of_hasArchCharacterAt4 below · cited by 2 · depth 18 - Lowest weight at a real place versus lowering annihilation
AutomorphicForm.isArchLowestWeightAt_iff_isArchLoweringAnnihilatedAt_of_hasArchCharacterAt3 below · cited by 2 · depth 18 - The adelic unipotent subgroup is closed in GL₂(A_K)
AutomorphicForm.isClosed_adelicUnipotent0 below · cited by 4 · depth 18 - Closedness of T(K)N(A_K) in GL₂(A_K)
AutomorphicForm.isClosed_rationalTorusUnipotent1 below · cited by 19 · depth 18 - Right convolution by a test function preserves vanishing constant term
AutomorphicForm.isCuspidalFn_rightConv4 below · cited by 13 · depth 18 - Normalised unipotent adelic measure is a bi-invariant Haar measure
AutomorphicForm.isHaarMeasure_and_isMulRightInvariant_unipotentHaar0 below · cited by 11 · depth 18 - Unimodular Haar measure on T(K)N(A_K)
AutomorphicForm.isHaarMeasure_rationalTorusUnipotentHaar_and_isMulRightInvariant4 below · cited by 19 · depth 18 - Isotypic cusp space at level bot vanishes
AutomorphicForm.isotypicCuspSubmodule_bot_eq_bot_of_productionPinsOf2 below · cited by 5 · depth 18 - Unfolding the Haar integral on GL₂(A_K) along N(A_K)
AutomorphicForm.lintegral_adelicGLHaar_eq_mul_lintegral_unipotentQuotientMeasure2 below · cited by 2 · depth 18 - Haar measure on GL₂(A_E) is σ-invariant
AutomorphicForm.measurePreserving_sigmaAdelicAct0 below · cited by 11 · depth 18 - Square-integrability of φ * f on a centre-cut Siegel window
AutomorphicForm.memLp_two_rightConv_restrict_of_isCuspAutomorphicFnAt_of_coversModCentre_of_pos75 below · cited by 4 · depth 18 - Cusp-realizable eigensystems are not Eisenstein tables
AutomorphicForm.not_agreesAwayFromFinite_twist_eisensteinTableOf_of_isArithGenuineCuspRealizable_of_coversModCentre211 below · cited by 1 · depth 18 - Translate package for right convolutions of cusp forms
AutomorphicForm.rightConv_translate_package_of_isCuspAutomorphicFnAt82 below · cited by 4 · depth 18 - Jensen bound for averaging over the archimedean maximal compact
AutomorphicForm.setLIntegral_nnnorm_integral_maximalCompactAtHaar_mul_sq_le_of_isFundamentalDomain11 below · cited by 2 · depth 18 - Bessel's inequality for Whittaker coefficients on the adelic box
AutomorphicForm.sum_norm_whittakerCoefficient_sq_le_integral_norm_sq1 below · cited by 6 · depth 18 - Non-vanishing of the first Whittaker coefficient over ℚ
AutomorphicForm.whittakerCoefficient_one_ne_zero_of_isIsotypicCuspFormAt_of_ne_zero10 below · cited by 8 · depth 18 - Unipotent Schwartz averaging multiplies the zeta integrand by int Bψ
AutomorphicForm.zetaIntegrand_whittakerCoefficient_unipotentAverage_eq_mul6 below · cited by 1 · depth 18 - Adjointness relations for Hecke eigenvalues under a covariant pairing
AutomorphicForm.a_mul_conj_b_eq_and_norm_b_eq_of_sesqForm_covariant_of_ne_zero4 below · cited by 4 · depth 19 - Adjointness of right convolution for the weighted Petersson pairing
AutomorphicForm.adjoint_rightConv_weightedPairing_of_isLsXiFunction10 below · cited by 12 · depth 19 - Archimedean Casimir commutes with right translation by GL₂(ℝ)
AutomorphicForm.archCasimirAt_comp_mul_archRealGLAt0 below · cited by 1 · depth 19 - Casimir at a real place via raising and lowering operators
AutomorphicForm.archCasimirAt_eq_raising_lowering_of_isArchSmoothAt1 below · cited by 1 · depth 19 - Casimir eigenvalue persists under right convolution by a test function
AutomorphicForm.archCasimirAt_rightConv_eq_smul_of_archCasimirAt_eq_smul_of_isArchSmoothAt_of_isFactorizableTestFn8 below · cited by 5 · depth 19 - Infinitesimal weight in along the rotation direction E-F
AutomorphicForm.archDerivAt_E_sub_archDerivAt_Fm_eq_smul_of_hasArchCharacterAt0 below · cited by 14 · depth 19 - mathfraksl₂ commutation relations for right-flow derivatives at a real place
AutomorphicForm.archDerivAt_commutator_of_isArchSmoothAt0 below · cited by 4 · depth 19 - Smoothing and integration by parts for right convolution
AutomorphicForm.archDerivAt_rightConv_eq_rightConv_deriv_of_isFactorizableTestFn3 below · cited by 23 · depth 19 - Raising a nonnegative real weight by two in an occurrence class
AutomorphicForm.archOccursInClassOf_archWeightChar_add_two_of_nonneg_of_coversModCentre147 below · cited by 2 · depth 19 - Weight symmetry n↦-n at a real place
AutomorphicForm.archOccursInClassOf_archWeightChar_neg_of_coversModCentre81 below · cited by 4 · depth 19 - Lowering annihilation at the lowest occurring weight
AutomorphicForm.archOccursInClassOf_isArchLoweringAnnihilatedAt_of_not_archOccursInClassOf_archWeightChar_sub_two_of_coversModCentre87 below · cited by 3 · depth 19 - Smoothness from holomorphic Iwasawa descent in weight k
AutomorphicForm.contDiffAt_of_mdifferentiable_im_cpow_mul_of_weight_of_central0 below · cited by 1 · depth 19 - C² regularity along the unipotent archimedean direction over ℚ
AutomorphicForm.contDiff_apply_unipotentGL2_mixedSpace_mul_of_isArchSmoothAt_rat0 below · cited by 1 · depth 19 - Continuity of unipotent Schwartz–Bruhat averages on GL₂(A_F)
AutomorphicForm.continuous_unipotentAverage6 below · cited by 1 · depth 19 - Convolution preserves the type-cut isotypic cusp space
AutomorphicForm.convOp_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_isUnitFactorizableAboveOfType71 below · cited by 3 · depth 19 - Parity of archimedean weights at a real place is constant
AutomorphicForm.even_sub_of_archOccursInClassOf_archWeightChar5 below · cited by 2 · depth 19 - Holomorphy of the Rankin–Selberg slab integral over a centre-cut Siegel cover
AutomorphicForm.exists_analyticOnNhd_eq_sub_mul_peterssonIntegral_of_norm_le_archHeight_pow_centreCutSiegelSet3 below · cited by 2 · depth 19 - Analyticity of a kernel-twisted Petersson integral on GL₂
AutomorphicForm.exists_analyticOnNhd_eq_sub_one_half_mul_peterssonIntegral_of_norm_le_archHeight_pow3 below · cited by 1 · depth 19 - Regularised Bruhat–Eisenstein family: continuation and moderate growth
AutomorphicForm.exists_analyticOnNhd_sub_one_half_mul_bruhatEisenstein_norm_le_archHeight_pow_of_isArchKFinite_family123 below · cited by 3 · depth 19 - Occurrence of a real-place weight character in Theta's class
AutomorphicForm.exists_archOccursInClassOf_archWeightChar_of_coversModCentre89 below · cited by 2 · depth 19 - Occurrence of a weight character at a real place
AutomorphicForm.exists_archOccursInClassOf_archWeightChar_of_coversModCentre_of_pos77 below · cited by 2 · depth 19 - Approximate identity with a single finite test factor
AutomorphicForm.exists_finTestFactor_isUnitFactorizableAboveOfType_tendsto_rightConv_of_mem_archCutSubmodule1 below · cited by 10 · depth 19 - Simultaneous splitting of the finite Whittaker factor over T
AutomorphicForm.exists_finWhittaker_eq_sum_prod_mul_of_isIsotypicCuspFormAt_placeEmbed_invariant_of_localSpaceAt14 below · cited by 4 · depth 19 - Composing convolution operators on isotypic cusp forms
AutomorphicForm.exists_finset_convOp_convOp_eq_sum_on_isotypicCuspSubmodule_inf_archCutSubmodule6 below · cited by 1 · depth 19 - Cyclicity of the isotypic cusp space under right convolution
AutomorphicForm.exists_finset_convOp_eq_of_ne_zero_of_mem_isotypicCuspSubmodule_inf_archCutSubmodule339 below · cited by 1 · depth 19 - Uniform decay of the first Whittaker coefficient along the torus
AutomorphicForm.exists_forall_prod_norm_pow_mul_norm_whittakerCoefficient_one_diagOne_unipotentAverage_le89 below · cited by 1 · depth 19 - Uniform square-integral bound over unipotent translates high in a Siegel set
AutomorphicForm.exists_forall_setLIntegral_nnnorm_sq_le_mul_archHeight_pow_mul_setLIntegral_of_isLsXiFunction_of_coversModCentre5 below · cited by 2 · depth 19 - Adelic GL₂ Haar measure against a compact open level
AutomorphicForm.exists_integral_archEntries_mul_indicator_eq_mul_integral_of_isCompact_of_isOpen0 below · cited by 3 · depth 19 - Archimedean bi-finiteness of a fibre integral on GL₂
AutomorphicForm.exists_isArchFactorBiFinite_of_forall_eq_integral_snoc0 below · cited by 1 · depth 19 - Bi-finitisation of the archimedean factor of a test function
AutomorphicForm.exists_isArchFactorBiFinite_rightConv_ne_zero_and_norm_sub_le_of_isCompact1 below · cited by 1 · depth 19 - Archimedean transfer for quadratic base change of GL₂
AutomorphicForm.exists_isArchTestFactor_forall_exists_isTwistedOrbitalIntegralOn_of_isEmpty_algHom46 below · cited by 1 · depth 19 - Archimedean approximate identity of prescribed archimedean types
AutomorphicForm.exists_isArchTestFactor_isArchFactorBiFinite_tendsto_integral_of_mem_archCutSubmodule0 below · cited by 4 · depth 19 - Properness of twisted conjugation modulo the twisted centraliser
AutomorphicForm.exists_isCompact_forall_sigmaConj_mem_exists_twistedCentralizer_mul3 below · cited by 7 · depth 19 - Replacing a test function by an archimedean bi-finite one
AutomorphicForm.exists_isFactorizableTestFn_isArchBiFinite_rightConv_eq_smul_of_rightConv_eq_smul9 below · cited by 4 · depth 19 - Irreducible GL₂(ℝ) K-type module with lowest weight k
AutomorphicForm.exists_isIrreducibleGL2RealKTypeModule_lower_eq_zero_iff_of_one_le0 below · cited by 1 · depth 19 - Existence of irreducible K-type modules of each parity
AutomorphicForm.exists_isIrreducibleGL2RealKTypeModule_ne_bot_iff_even_sub0 below · cited by 1 · depth 19 - Continuous section functions at regular semisimple γ over K_∞
AutomorphicForm.exists_isSectionFnOn_infiniteAdeleRing_and_continuous_of_isRegularSemisimple_of_hasCompactSupport1 below · cited by 6 · depth 19 - Locally constant section functions at regular semisimple elements of GL₂(Kᵥ)
AutomorphicForm.exists_isSectionFn_and_isLocallyConstant_of_isRegularSemisimple_of_isLocalTestFn1 below · cited by 7 · depth 19 - Nonzero isotypic cusp form invariant under a level-one subgroup
AutomorphicForm.exists_levelOne_invariant_isIsotypicCuspFormAt_principalLevel_ne_zero_of_ne_zero96 below · cited by 1 · depth 19 - An equivariant archimedean type projector on K-finite functions
AutomorphicForm.exists_linearMap_archCutProjector_comm_rightTranslate2 below · cited by 3 · depth 19 - Cyclic prime-degree base change from a non-vanishing twisted cut trace
AutomorphicForm.exists_mem_cuspClasses_principalLevel_of_twistedCutTrace_ne_zero_of_areMatchingAt_inv_of_prime3,062 below · cited by 1 · depth 19 - Non-vanishing Whittaker coefficient at a principal idele
AutomorphicForm.exists_mem_principalIdeles_whittakerCoefficient_one_diagOne_mul_ne_zero24 below · cited by 1 · depth 19 - Local transfer near δ₀ with regular semisimple norm string
AutomorphicForm.exists_nhds_forall_exists_isLocalTestFn_areMatchingLocal_of_isRegularSemisimple_normString7 below · cited by 1 · depth 19 - Local transfer near an element with non-regular norm string
AutomorphicForm.exists_nhds_forall_exists_isLocalTestFn_areMatchingLocal_of_not_isRegularSemisimple_normString21 below · cited by 1 · depth 19 - Decay of a convolved cusp form along diag(a,1)
AutomorphicForm.exists_norm_rightConv_diagOne_mul_mul_unipotentGL2_le_of_le_ideleNorm89 below · cited by 1 · depth 19 - Reproduction and Whittaker properties of isotypic cusp forms over ℚ
AutomorphicForm.exists_rightConv_eq_self_and_isIsotypicCuspFormAt_add_smul_archDerivAt_and_whittakerCoefficient_bounds_of_mem_archCutSubmodule351 below · cited by 1 · depth 19 - Pole at s=1/2 of the Bruhat–Eisenstein family on GL₂
AutomorphicForm.exists_tendsto_sub_one_half_mul_bruhatEisenstein_continuation_of_isArchKFinite_family180 below · cited by 3 · depth 19 - Support of the first Whittaker coefficient on the torus diag(b,1)
AutomorphicForm.exists_whittakerCoefficient_one_diagOne_eq_zero_of_exp_lt_valuation24 below · cited by 3 · depth 19 - Nonvanishing Whittaker coefficient at a diagonal point over ℚ
AutomorphicForm.exists_whittakerCoefficient_one_diagOne_ne_zero_of_glFin_eq_one_rat2 below · cited by 1 · depth 19 - Finite-dimensionality of convolution-fixed adelic cusp forms
AutomorphicForm.finiteDimensional_of_forall_mem_rightConv_eq_self75 below · cited by 1 · depth 19 - Vectors in an archimedean cut are K_∞¹-finite
AutomorphicForm.finiteDimensional_span_rightTranslate_of_mem_archCutSubmodule0 below · cited by 3 · depth 19 - Differentiating right convolution along archimedean unipotent directions
AutomorphicForm.hasDerivAt_rightConv_mul_unipotentGL2_and_isFactorizableTestFn_leftDeriv_and_linear2 below · cited by 2 · depth 19 - Archimedean derivatives and Casimir pass through Whittaker coefficients
AutomorphicForm.hasDerivAt_whittakerCoefficient_archFlow_of_continuous_archDerivAt0 below · cited by 3 · depth 19 - Sphericity and Hecke eigenvalue survive right convolution
AutomorphicForm.heckeCosetSum_sum_rightConv_translate_eq_of_pure_reps1 below · cited by 5 · depth 19 - Independence of the weighted integral from the section weight
AutomorphicForm.integral_mul_eq_integral_mul_of_forall_integral_subgroup_mul_eq_one0 below · cited by 9 · depth 19 - Shear change of variables for twisted orbital integrals
AutomorphicForm.integral_twistedConj_mul_eq_integral_conj_fibreIntegral_mul0 below · cited by 4 · depth 19 - Lowering annihilation at a real place: slices versus flow derivatives
AutomorphicForm.isArchLoweringAnnihilatedAt_iff_isArchSmoothAt_and_lower_eq_zero_of_hasArchCharacterAt9 below · cited by 4 · depth 19 - Archimedean smoothness from holomorphy of Iwasawa descents
AutomorphicForm.isArchSmoothAt_of_mdifferentiable_cpow_mul_descent_of_hasArchCharacterAt0 below · cited by 1 · depth 19 - Reflected lowering operator: weight one, Casimir eigenvalue, T²=1-4λ
AutomorphicForm.isArchSmoothAt_reflectedLowering_and_archCasimirAt_eq_and_reflectedLowering_reflectedLowering_eq_smul0 below · cited by 3 · depth 19 - Two-sided χ-averaging of an archimedean test factor
AutomorphicForm.isArchTestFactor_and_isArchFactorBiFinite_ofChar_integral_of_isArchTestFactor0 below · cited by 1 · depth 19 - Fibre integrals along multiplication give archimedean test factors
AutomorphicForm.isArchTestFactor_of_forall_eq_integral_snoc0 below · cited by 1 · depth 19 - Compactness of the level group with trivial archimedean part
AutomorphicForm.isCompact_levelOne_inf_finiteAdelicGL2Subgroup0 below · cited by 14 · depth 19 - Averaging over the maximal compact preserves cuspidality
AutomorphicForm.isCuspidalFn_integral_maximalCompactAtHaar_mul_of_isCuspidalFn0 below · cited by 1 · depth 19 - Schwartz–Bruhat unipotent averages of cuspidal functions are cuspidal
AutomorphicForm.isCuspidalFn_unipotentAverage3 below · cited by 1 · depth 19 - Factorizable test functions are stable under twisting by η∘det
AutomorphicForm.isFactorizableTestFn_chiDet_mul_of_continuous_of_isOfFinOrder0 below · cited by 1 · depth 19 - Right convolution preserves the isotypic cusp space
AutomorphicForm.isIsotypicCuspFormAt_rightConv_of_isFactorizableTestFn_of_support_subset_of_coversModCentre79 below · cited by 9 · depth 19 - Local double-coset sums preserve isotypic cusp forms
AutomorphicForm.isIsotypicCuspFormAt_sum_apply_mul_finEmbed_localEmbed_of_isHeckeCosetSystem23 below · cited by 1 · depth 19 - Right convolution by a factorizable test function is K_f-smooth
AutomorphicForm.isKfSmooth_rightConv1 below · cited by 21 · depth 19 - K_f-smoothness of unipotent Schwartz–Bruhat averages
AutomorphicForm.isKfSmooth_unipotentAverage0 below · cited by 2 · depth 19 - Flat involution preserves level-spherical functions of unitary character type
AutomorphicForm.isLevelSphericalOfType_ofChar_flat6 below · cited by 1 · depth 19 - Twisting a smooth cusp form by a finite-order Hecke character
AutomorphicForm.isSmoothCuspAutomorphicFnAt_twistedCentralChar_fnTwist_productionPinsOf0 below · cited by 1 · depth 19 - Vanishing of isotypic cusp spaces when the height floor is non-positive
AutomorphicForm.isotypicCuspSubmodule_eq_bot_of_nonpos5 below · cited by 2 · depth 19 - From a slab fundamental domain to centre-cut Siegel windows
AutomorphicForm.isotypicCuspSubmodule_inf_archCutSubmodule_le_of_isFundamentalDomain_of_pos336 below · cited by 3 · depth 19 - Isotypic cusp spaces shrink when the determinant floor is lowered
AutomorphicForm.isotypicCuspSubmodule_le_isotypicCuspSubmodule_of_le_of_ne_bot4 below · cited by 1 · depth 19 - Maass raising and lowering operators at a real place
AutomorphicForm.iterate_raise_iterate_lower_eq_smul_of_archCasimirAt_eq_smul0 below · cited by 6 · depth 19 - Lowering operator L=(H-iS)/2 drops the SO(2)-weight by two
AutomorphicForm.lowering_fderiv_mul_rotation_eq_exp_mul_of_weight0 below · cited by 2 · depth 19 - Holomorphy of y^σ-descents versus the lowering operator
AutomorphicForm.mdifferentiable_cpow_mul_descent_iff_lower_eq_smul_of_isArchSmoothAt0 below · cited by 1 · depth 19 - Holomorphy of y^σF versus the lowering eigenvalue equation
AutomorphicForm.mdifferentiable_im_cpow_mul_iff_forall_lowering_fderiv_eq0 below · cited by 1 · depth 19 - Measurability of unipotent integrals on N(A_K)backslashGL₂(A_K)
AutomorphicForm.measurable_lintegral_unipotentGL2_mul_out2 below · cited by 4 · depth 19 - Convolution eigenvectors of isotypic cusp forms are K-finite
AutomorphicForm.mem_cuspKFiniteSubmodule_of_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_rightConv_eq_smul86 below · cited by 1 · depth 19 - Archimedean type characters of a nonzero continuous form are unitary
AutomorphicForm.norm_archChar_eq_one_of_mem_archCutSubmodule_ofChar1 below · cited by 1 · depth 19 - No weight k-2 beneath a lowering-annihilated weight k
AutomorphicForm.not_archOccursInClassOf_archWeightChar_sub_two_of_isArchLoweringAnnihilatedAt_of_coversModCentre387 below · cited by 2 · depth 19 - Lowering-annihilated cusp forms at a real place have weight k ≥ 1
AutomorphicForm.one_le_of_archOccursInClassOf_isArchLoweringAnnihilatedAt_of_coversModCentre136 below · cited by 4 · depth 19 - Rankin–Selberg unfolding on a determinant slab for GL₂
AutomorphicForm.peterssonIntegral_mul_bruhatEisenstein_eq_integral_whittakerCoefficient_mul_conj_rationalCentreUnipotentQuotient38 below · cited by 2 · depth 19 - Non-vanishing of the self-Petersson integral over a slab fundamental domain
AutomorphicForm.peterssonIntegral_self_ne_zero_of_isFundamentalDomain_of_continuous6 below · cited by 8 · depth 19 - Right convolution by an arch-bi-finite test function stays in the cut
AutomorphicForm.rightConv_mem_archCutSubmodule_of_isArchBiFinite2 below · cited by 25 · depth 19 - Associativity of right convolution on GL₂(A_F)
AutomorphicForm.rightConv_rightConv_eq_rightConv_rightConv_comp_inv2 below · cited by 1 · depth 19 - Right translation by archimedean row isometries commutes with smoothing
AutomorphicForm.rightTranslate_rightConv_of_isLevelSphericalOfType2 below · cited by 5 · depth 19 - Casimir symmetry and raising–lowering adjointness on a fundamental domain
AutomorphicForm.setIntegral_archCasimirAt_mul_conj_eq_and_lower_adjoint_of_isFundamentalDomain19 below · cited by 6 · depth 19 - Unfolding along the unipotent subgroup of adelic GL₂
AutomorphicForm.setLIntegral_adelicGLHaar_eq_lintegral_unipotentQuotientMeasure2 below · cited by 4 · depth 19 - First-moment bound sumₚ |aₚ| Np^{-σ}<∞ for σ>1
AutomorphicForm.summable_norm_a_mul_rpow_absNorm_of_isArithGenuineCuspRealizable715 below · cited by 1 · depth 19 - Type pieces of inequivalent irreducibles meet in zero
AutomorphicForm.typeSubmodule_inf_typeSubmodule_eq_bot0 below · cited by 2 · depth 19 - Left GL₂(F)-invariance of unipotent averages
AutomorphicForm.unipotentAverage_globalPoints_mul0 below · cited by 2 · depth 19 - Integrability and summability of adelic Whittaker coefficients over ℚ
AutomorphicForm.whittakerCoefficientIntegrable_and_summable_of_isKfSmooth_of_contDiff12 below · cited by 7 · depth 19 - Vanishing of the Whittaker coefficient at g Gᵥ^{-(k+1)}
AutomorphicForm.whittakerCoefficient_mul_heckeGen_pow_inv_eq_zero1 below · cited by 1 · depth 19 - Whittaker coefficient: Hecke representatives raise the exponent
AutomorphicForm.whittakerCoefficient_mul_heckeGen_pow_mul_localRepSome_eq1 below · cited by 1 · depth 19 - Central step-down of Whittaker coefficients along Hecke powers
AutomorphicForm.whittakerCoefficient_mul_heckeGen_pow_succ_mul_localRepInf_eq0 below · cited by 1 · depth 19 - Linearity of the adelic-box Whittaker coefficient in φ
AutomorphicForm.whittakerCoefficient_sum_smul_of_continuous0 below · cited by 17 · depth 19 - Archimedean Whittaker coefficient: covariance, ODE, growth, separation
AutomorphicForm.whittakerCoefficient_torus_peel_ode_growth_and_separation_of_isIsotypicCuspFormAt_of_archCasimirAt_eq_smul10 below · cited by 1 · depth 19 - Whittaker coefficients of a unipotent average at diag(a,1)
AutomorphicForm.whittakerCoefficient_unipotentAverage_diagOne5 below · cited by 3 · depth 19 - Agreement of Hecke eigensystems from an invariant pairing
AutomorphicForm.agreesAwayFromFinite_of_projInvariant_sesqForm_ne_zero1 below · cited by 1 · depth 20 - Vanishing unipotent derivatives force right SL₂(ℝ)-invariance at a real place
AutomorphicForm.apply_mul_archRealGLAt_eq_of_archDerivAt_E_eq_zero_of_archDerivAt_Fm_eq_zero0 below · cited by 3 · depth 20 - Smoothness and sphericity outside S give right K^S-invariance
AutomorphicForm.apply_mul_eq_of_isKfSmooth_of_forall_placeEmbed_of_mem_maximalCompactAway1 below · cited by 3 · depth 20 - Weight-k forms satisfy (E-F)φ = ik φ at a real place
AutomorphicForm.archDerivAt_E_sub_archDerivAt_Fm_eq_smul_of_hasArchCharacterAtZero_of_isArchSmoothAt0 below · cited by 3 · depth 20 - J-rigidity of weight-one class witnesses over ℚ
AutomorphicForm.archOccursInClassOf_archWeightChar_one_apply_mul_archRealGLAt_J_eq_mul_lower_of_ne_of_coversModCentre_rat370 below · cited by 2 · depth 20 - Weight-zero occurrence can be taken J-eigen at a real place
AutomorphicForm.archOccursInClassOf_archWeightChar_zero_apply_mul_archRealGLAt_J_eq_of_coversModCentre10 below · cited by 2 · depth 20 - Shell-boundedness of derivatives of a Casimir eigen-witness at a real place
AutomorphicForm.archOccursInClassOf_continuous_foldr_archDerivAt_of_archOccursInClassOf_archCasimirAt_eq_smul_of_coversModCentre95 below · cited by 3 · depth 20 - Occurrence in a Hecke class is independent of the determinant floor
AutomorphicForm.archOccursInClassOf_iff_archOccursInClassOf_of_le_of_pos_of_coversModCentre8 below · cited by 1 · depth 20 - Lowering-annihilated weight-k witness when weight k-2 is absent
AutomorphicForm.archOccursInClassOf_isArchLoweringAnnihilatedAt_of_not_archOccursInClassOf_archWeightChar_sub_two_of_coversModCentre_of_pos76 below · cited by 1 · depth 20 - Raising operator kills weight-k forms with extremal Casimir eigenvalue
AutomorphicForm.archOccursInClassOf_raise_eq_zero_of_archCasimirAt_eq_smul_of_coversModCentre24 below · cited by 1 · depth 20 - Unit fundamental lemma at unramified v, prime degree
AutomorphicForm.areMatchingLocal_indicator_semiLocalIntegralSet_of_ramificationIdx_eq_one_of_prime78 below · cited by 3 · depth 20 - Spherical fundamental lemma at a split place, prime degree
AutomorphicForm.areMatchingLocal_splitFactor_heckeAlgebra_of_prime4 below · cited by 3 · depth 20 - Holomorphy of the Bruhat Eisenstein series for an entire family
AutomorphicForm.bruhatEisenstein_differentiableOn_re_gt_half_of_entire_family5 below · cited by 5 · depth 20 - Fourier–Whittaker expansion of the Bruhat Eisenstein series
AutomorphicForm.bruhatEisenstein_eq_constantTerm_add_whittakerSum_of_one_lt_re_of_unitary33 below · cited by 5 · depth 20 - Summability of the big-cell Bruhat summand for Re s>1/2
AutomorphicForm.bruhatTransversal_summand_norm_summable_of_re_gt_half0 below · cited by 22 · depth 20 - Additivity of the constant-term integral
AutomorphicForm.constantTerm_add0 below · cited by 2 · depth 20 - Constant term of the Bruhat Eisenstein series for Re s>1/2
AutomorphicForm.constantTerm_bruhatEisenstein_eq_section_add_weylIntertwiningIntegral1 below · cited by 20 · depth 20 - Homogeneity of the constant term in the function
AutomorphicForm.constantTerm_smul0 below · cited by 1 · depth 20 - Continuity of right convolution on adelic GL₂
AutomorphicForm.continuous_rightConv_of_continuous_of_hasCompactSupport2 below · cited by 9 · depth 20 - Convolution of factorizable test functions on adelic GL₂
AutomorphicForm.convOp_convOp_eq_convOp_of_eq_integral_mul_comp_inv_mul5 below · cited by 3 · depth 20 - Hecke coset sums commute with right convolution by spherical f
AutomorphicForm.cosetSum_rightConv_of_isLevelSphericalOfType1 below · cited by 2 · depth 20 - Global additive character with e^{2π i t} at a real place is standard
AutomorphicForm.eq_stdAddChar_of_isGlobalAddChar_of_apply_infinitePlace_eq_exp12 below · cited by 4 · depth 20 - Vanishing on a torus ray kills a determinant component
AutomorphicForm.eq_zero_of_forall_torusRay_eq_zero_of_mul_det_pos0 below · cited by 1 · depth 20 - Cuspidal automorphic functions vanishing on a covering window vanish
AutomorphicForm.eq_zero_of_isCuspAutomorphicFnAt_productionPinsOf_of_coversModCentre_of_forall_mem_eq_zero1 below · cited by 1 · depth 20 - Cuspidal functions trivial under SL₂(ℝ) at a real place vanish
AutomorphicForm.eq_zero_of_isCuspidalFn_of_forall_apply_mul_archRealGLAt_eq10 below · cited by 1 · depth 20 - Vanishing of L² ξ-automorphic functions when c≤ 0
AutomorphicForm.eq_zero_of_isLsXiFunction_of_memLp_of_nonpos_of_coversModCentre0 below · cited by 2 · depth 20 - Bruhat representatives for Bbackslash GL₂(K)
AutomorphicForm.existsUnique_bruhatRepresentative_mul_mem_borelSubgroup0 below · cited by 3 · depth 20 - Unique diag(a,1) representative for B(K) modulo Z(K)N(K)
AutomorphicForm.existsUnique_diagOne_inv_mul_mem_scalar_sup_unipotent_of_mem_borelSubgroup0 below · cited by 3 · depth 20 - Analytic continuation and rapid decay of the non-constant part of Eₛ
AutomorphicForm.exists_analyticOnNhd_bruhatEisenstein_sub_constantTerm_norm_le_rpow_neg_of_isArchKFinite_family121 below · cited by 1 · depth 20 - Regularised Weyl intertwining integral continues past Re s=1/2
AutomorphicForm.exists_analyticOnNhd_sub_one_half_mul_weylIntertwiningIntegral_isInducedSection_of_isArchKFinite_family65 below · cited by 3 · depth 20 - Continuous central idempotent reproducing a finite-dimensional archimedean type space
AutomorphicForm.exists_continuous_conj_invariant_integral_mul_apply_mul_eq_of_finiteDimensional_of_le_archCutSubmodule1 below · cited by 6 · depth 20 - Depth of lower unipotent invariance for translated level-N vectors
AutomorphicForm.exists_depth_forall_apply_mul_lowerUnipotentGL2_eq_of_sum_translate0 below · cited by 2 · depth 20 - Non-zero vectors generate the level-and-type subspace of a cuspidal constituent
AutomorphicForm.exists_finset_convOp_eq_of_isCuspConstituent_of_ne_zero170 below · cited by 1 · depth 20 - Flat Eisenstein series bounded on centre-cut Siegel sets
AutomorphicForm.exists_flatEisenstein_mul_le_mul_archHeight_rpow_of_mem_centreCutSiegelSet5 below · cited by 5 · depth 20 - Flat entire family of induced sections through a given section
AutomorphicForm.exists_flat_isInducedSection_family_eq_of_isInducedSection4 below · cited by 15 · depth 20 - A single Casimir eigenvalue on every isotypic cut
AutomorphicForm.exists_forall_archCasimirAt_eq_smul_of_mem_isotypicCuspSubmodule_of_mem_archCutSubmodule_of_coversModCentre353 below · cited by 5 · depth 20 - Spherical base change matching at an inert place, prime degree
AutomorphicForm.exists_heckeAlgHom_areMatchingLocal_of_inert_of_prime70 below · cited by 3 · depth 20 - Centre-cut Siegel windows are neighbourhoods in adelic GL₂
AutomorphicForm.exists_iUnion_centreCutSiegelSet_mem_nhds0 below · cited by 1 · depth 20 - Archimedean bi-finiteness of convolution kernels on GL₂(A_F)
AutomorphicForm.exists_isArchBiFinite_rightConv_comp_inv0 below · cited by 1 · depth 20 - Archimedean transfer for ramified quadratic base change of GL₂
AutomorphicForm.exists_isArchTestFactor_forall_isNormConjugator_one_exists_isTwistedOrbitalIntegralOn_of_isEmpty_algHom45 below · cited by 1 · depth 20 - Borel-times-compact factorisation of right translates, with height bounds
AutomorphicForm.exists_isCompact_forall_mul_eq_borel_mul_archHeight_le_of_glFin_mem_finiteIntegralGL25 below · cited by 2 · depth 20 - Level-adapted test function preserving the archimedean type at w
AutomorphicForm.exists_isFactorizableTestFn_hasArchCharacterAt_rightConv_ne_zero_of_hasArchCharacterAt2 below · cited by 2 · depth 20 - One bi-finite test function reproduces a finite-dimensional cut
AutomorphicForm.exists_isFactorizableTestFn_isArchBiFinite_forall_rightConv_eq_self_of_finiteDimensional_of_isCompact22 below · cited by 1 · depth 20 - Eisenstein unfolding to a rational Borel fundamental domain
AutomorphicForm.exists_isFundamentalDomain_borel_setIntegral_eq_peterssonIntegral_mul_bruhatEisenstein13 below · cited by 1 · depth 20 - Transport of coupled twisted orbital data along σ-conjugation
AutomorphicForm.exists_isHaarMeasure_coupled_one_of_coupled_sigmaConjugate0 below · cited by 3 · depth 20 - Adelic spans embed equivariantly into copies of one irreducible representation
AutomorphicForm.exists_isIrreducibleGLRep_injective_linearMap_adelicSpan_finsupp_of_agreesAwayFromFinite341 below · cited by 1 · depth 20 - Weight-n projection of an isotypic cusp form at a real place
AutomorphicForm.exists_isIsotypicCuspFormAt_hasArchCharacterAt_whittakerCoefficient_eq_of_whittakerCoefficient_mul_archIncl_eq3 below · cited by 4 · depth 20 - Flat level-spherical approximate identity for a finite-dimensional space
AutomorphicForm.exists_isLevelSphericalOfType_flat_tendsto_rightConv_of_finiteDimensional19 below · cited by 1 · depth 20 - Local transfer near an anisotropic base point
AutomorphicForm.exists_isOpen_one_mem_forall_exists_isLocalTestFn_of_forall_mul_sigmaTensor_ne16 below · cited by 2 · depth 20 - Existence of local orbital integrals at regular semisimple elements
AutomorphicForm.exists_isOrbitalIntegral_of_isRegularSemisimple_of_isLocalTestFn1 below · cited by 9 · depth 20 - Right-equivariant maps into ℂ^G extend from subrepresentations
AutomorphicForm.exists_isRightEquivariant_comp_subtype_eq_of_injective0 below · cited by 8 · depth 20 - Continuous twisted sections at archimedean places
AutomorphicForm.exists_isTwistedSectionFnOn_infiniteAdeleRing_and_continuous_of_isRegularSemisimple_normString_of_hasCompactSupport0 below · cited by 6 · depth 20 - Bi-invariant unit-factorizable test function with non-zero convolution
AutomorphicForm.exists_isUnitFactorizableAboveOfType_biInvariant_rightConv_ne_zero_of_mem_archCutSubmodule4 below · cited by 2 · depth 20 - One-prime step towards U₁-invariance at principal level
AutomorphicForm.exists_levelOne_pow_invariant_isIsotypicCuspFormAt_principalLevel_ne_zero_of_ne_zero94 below · cited by 1 · depth 20 - Left Haar measure of the adelic Borel subgroup in coordinates
AutomorphicForm.exists_lintegral_adelicBorel_eq_mul_lintegral_coord_of_isMulLeftInvariant3 below · cited by 1 · depth 20 - Iwasawa disintegration of the Z(K)N(A)-quotient measure on GL₂
AutomorphicForm.exists_lintegral_rationalCentreUnipotentQuotientMeasure_eq_mul_setLIntegral_iwasawa14 below · cited by 2 · depth 20 - Fundamental domain in centre-cut Siegel translates over a determinant slab
AutomorphicForm.exists_measurableSet_isFundamentalDomain_subset_iUnion_centreCutSiegelSet_of_coversModCentre12 below · cited by 13 · depth 20 - Nonzero isotypic cusp form has nonzero archimedean-type component
AutomorphicForm.exists_mem_archCutSubmodule_isIsotypicCuspFormAt_ne_zero75 below · cited by 1 · depth 20 - Nonzero twisted cut trace carried by a fibre-constant cuspidal class
AutomorphicForm.exists_mem_cuspClasses_twistedCutTrace_ne_zero_of_twistedCutTrace_ne_zero_of_prime351 below · cited by 1 · depth 20 - Iwasawa normalisation of a non-vanishing Whittaker value
AutomorphicForm.exists_mem_maximalCompactAt_apply_diagOne_mul_ne_zero_of_apply_ne_zero2 below · cited by 2 · depth 20 - Archimedean derivative of a unipotent average's first Whittaker coefficient
AutomorphicForm.exists_mem_schwartzBruhat_whittakerCoefficient_unipotentAverage_diagOne_eq_trace_mul8 below · cited by 1 · depth 20 - Euler factors normalising the Weyl intertwining integral
AutomorphicForm.exists_meromorphicOn_partialEulerProduct_mul_weylIntertwiningIntegral_eq_mul56 below · cited by 10 · depth 20 - Compact open kernel in the centraliser of a regular semisimple γ
AutomorphicForm.exists_monoidHom_localCentralizer_isCompact_ker_of_isRegularSemisimple0 below · cited by 1 · depth 20 - Polynomial bound on Hecke eigenvalues of a cusp realization
AutomorphicForm.exists_norm_a_le_absNorm_rpow_and_norm_b_le_of_smoothCuspRealizationAt_of_peterssonPairing0 below · cited by 2 · depth 20 - Induced sections on GL₂(A_F) are bounded by H^{Res+1/2}
AutomorphicForm.exists_norm_le_mul_adelicHeight_rpow_of_isInducedSection5 below · cited by 4 · depth 20 - Rapid decay of φ * f in the adelic height on a determinant slab
AutomorphicForm.exists_norm_rightConv_le_mul_inv_adelicHeight_pow_of_ideleNorm_det_mem_Icc78 below · cited by 2 · depth 20 - Coordinatewise rapid decay of smoothed cuspidal Whittaker coefficients
AutomorphicForm.exists_norm_whittakerCoefficient_rightConv_diagOne_mul_le_ideleNorm_rpow_mul_norm_infinitePlace_rpow_neg92 below · cited by 7 · depth 20 - Sandwiched smoothing on reproduced vectors is archimedean smoothing
AutomorphicForm.exists_pos_forall_rightConv_integral_prod_mul_indicator_eq_mul_integral_of_forall_integral_mul_apply_mul_eq4 below · cited by 5 · depth 20 - Transporting a continuous cusp realization to the standard Siegel window
AutomorphicForm.exists_smoothCuspRealizationAt_productionPinsGeneral_toFun_eq_of_coversModCentre21 below · cited by 6 · depth 20 - Non-zero g-independent limit of the normalised intertwining integral
AutomorphicForm.exists_tendsto_tprod_one_sub_absNorm_cpow_mul_weylIntertwiningIntegral_nhds_one_half_of_isArchKFinite_family101 below · cited by 1 · depth 20 - Unipotent surgery cutting Whittaker support to the units
AutomorphicForm.exists_unipotent_surgery_whittakerCoefficient_diagOne_mul_eq_sum_mul9 below · cited by 1 · depth 20 - Unipotent surgery cutting a Whittaker function to a valuation shell
AutomorphicForm.exists_unipotent_surgery_whittakerCoefficient_diagOne_mul_eq_sum_mul_shell9 below · cited by 1 · depth 20 - Euler product for Whittaker sums of GL₂ Eisenstein families
AutomorphicForm.exists_unitaryChar_entire_partialEulerProduct_mul_eq_tsum_whittakerCoefficient_bruhatEisenstein64 below · cited by 1 · depth 20 - Fibrewise twisted trace comparison at prime degree
AutomorphicForm.fibreSum_twistedCutTrace_eq_const_mul_fibreSum_cutTrace_of_areMatchingAt_symm_of_prime3,006 below · cited by 1 · depth 20 - Finite-dimensional stable span of archimedean translates
AutomorphicForm.finiteDimensional_span_translates_of_mem_archCutSubmodule1 below · cited by 7 · depth 20 - Dichotomy for type-cut isotypic cusp spaces over a covering Siegel window
AutomorphicForm.forall_isotypicCuspSubmodule_inf_archCutSubmodule_eq_bot_or_forall_eq_of_coversModCentre338 below · cited by 2 · depth 20 - Whittaker expansion over principal ideles of a cuspidal function
AutomorphicForm.hasSum_whittakerCoefficient_one_diagOne_principalIdeles_mul23 below · cited by 3 · depth 20 - Bargmann's bound for Casimir eigenvalues of class witnesses
AutomorphicForm.im_eq_zero_and_le_re_of_archOccursInClassOf_archCasimirAt_eq_smul_of_coversModCentre24 below · cited by 3 · depth 20 - Casimir scalar at a real place: reality, positivity, weight formula
AutomorphicForm.im_eq_zero_and_re_pos_and_eq_of_forall_archCasimirAt_eq_of_coversModCentre388 below · cited by 1 · depth 20 - Integrability of ‖φ*f‖² against height powers on Siegel pieces
AutomorphicForm.integrableOn_norm_rightConv_sq_mul_archHeight_pow_mul_ideleNorm_rpow_inter_centreCutSiegelSet88 below · cited by 3 · depth 20 - Parseval identity for Whittaker coefficients on the adelic box
AutomorphicForm.integral_mul_conj_eq_tsum_whittakerCoefficient_mul_conj6 below · cited by 3 · depth 20 - Parseval step of Rankin–Selberg unfolding over the rational torus
AutomorphicForm.integral_mul_conj_unipotent_eq_tsum_units_whittakerCoefficient_one_diagOne_and_tsum_norm_le12 below · cited by 2 · depth 20 - Two-sided average of a conjugation-invariant flat function
AutomorphicForm.integral_prod_conj_eq_and_eq_conj_mul_of_conj_invariant_of_flat0 below · cited by 5 · depth 20 - Rankin–Selberg unfolding along the rational torus on GL₂
AutomorphicForm.integral_rationalTorusUnipotentQuotient_tsum_units_eq_integral_rationalCentreUnipotentQuotient17 below · cited by 2 · depth 20 - Two-sided K-averages of archimedean test factors
AutomorphicForm.isArchTestFactor_and_isArchFactorBiFinite_integral_prod_of_continuous_of_mem_iSup_typeSubmodule1 below · cited by 6 · depth 20 - Right convolution preserves smooth compactly supported archimedean factors
AutomorphicForm.isArchTestFactor_integral_mul_of_isArchTestFactor_of_hasCompactSupport0 below · cited by 1 · depth 20 - Boundedness of smoothed cusp forms on Siegel windows
AutomorphicForm.isBoundedOnSiegelWindows_rightConv_of_isCuspAutomorphicFnAt_of_isFundamentalDomain73 below · cited by 2 · depth 20 - Z(K)N(A_K) is closed in GL₂(A_K)
AutomorphicForm.isClosed_rationalCentreUnipotent1 below · cited by 7 · depth 20 - Compactness of the determinant-one row isometry group at w
AutomorphicForm.isCompact_rowIsometrySubgroup_detOne0 below · cited by 18 · depth 20 - Right convolution preserves finite test factors
AutomorphicForm.isFinTestFactor_integral_mul_of_isFinTestFactor_of_hasCompactSupport1 below · cited by 1 · depth 20 - Box sheet as fundamental domain for Z(K)N(K)
AutomorphicForm.isFundamentalDomain_boxSheet_rationalCentreUnipotent0 below · cited by 3 · depth 20 - Box sheet is a fundamental domain for B(K)
AutomorphicForm.isFundamentalDomain_boxSheet_rationalTorusUnipotent0 below · cited by 15 · depth 20 - Z(K)N(A_K)-measure is a left and right invariant Haar measure
AutomorphicForm.isHaarMeasure_rationalCentreUnipotentHaar_and_isMulRightInvariant1 below · cited by 7 · depth 20 - Explicit induced section on adelic GL₂ with prescribed level
AutomorphicForm.isInducedSection_indicator_bottomRow_mul_adelicHeight_cpow4 below · cited by 3 · depth 20 - Isotypic cusp forms transfer to ample Siegel windows
AutomorphicForm.isIsotypicCuspFormAt_centreCutSiegelSetAmple_of_isIsotypicCuspFormAt_of_coversModCentre20 below · cited by 2 · depth 20 - Local constancy and measurability of the Iwasawa shell index
AutomorphicForm.isLocallyConstant_iwasawaShellIndex_and_measurable0 below · cited by 2 · depth 20 - Left invariance of the Iwasawa shell index under Z(K)N(A)
AutomorphicForm.iwasawaShellIndex_mul_of_mem_rationalCentreUnipotent0 below · cited by 2 · depth 20 - Left and right Casimir agree at a real place
AutomorphicForm.leftCasimir_eq_archCasimirAt_of_isArchSmoothAt0 below · cited by 5 · depth 20 - Conjugation by the Hecke element scales the Z(K)N measure by Nv
AutomorphicForm.lintegral_rationalCentreUnipotentHaar_comp_heckeGen_mul_centralScalar_conj3 below · cited by 1 · depth 20 - L²-ness of window-bounded forms on covering centre-cut Siegel windows
AutomorphicForm.memLp_two_of_isBoundedOnSiegelWindows_of_exists_memLp_two_of_coversModCentre6 below · cited by 3 · depth 20 - Conjugation by diag(varpiᵥ,1)z(u) preserves Z(K)N(A)
AutomorphicForm.mem_rationalCentreUnipotent_iff_heckeGen_mul_centralScalar_conj_mem0 below · cited by 1 · depth 20 - Level-M² invariance of a twisted translate of φ
AutomorphicForm.mul_dirichletIdeleChar_det_rightTranslate_invariant_levelOne_sq0 below · cited by 1 · depth 20 - Covering by centre-cut Siegel windows forces u ≠ 0
AutomorphicForm.ne_zero_of_coversModCentre_iUnion_centreCutSiegelSet1 below · cited by 2 · depth 20 - Commuting level-spherical and bi-finite right convolutions on GL₂
AutomorphicForm.rightConv_rightConv_comm_of_isLevelSphericalOfType1 below · cited by 1 · depth 20 - Cuspidal subrepresentations absorb smoothings of arbitrary right translates
AutomorphicForm.rightConv_rightTranslate_mem_of_isCuspSubrep4 below · cited by 2 · depth 20 - Right translation is adjoint for the weighted Petersson pairing
AutomorphicForm.rightTranslate_adjoint_weightedPairing_of_isLsXiFunction9 below · cited by 10 · depth 20 - Skew-symmetry of real-place flow derivatives on a determinant slab
AutomorphicForm.setIntegral_archDerivAt_mul_conj_add_eq_zero_of_isFundamentalDomain17 below · cited by 2 · depth 20 - Rankin–Selberg second moment for cusp-realizable eigensystems over ℚ
AutomorphicForm.summable_norm_a_sq_mul_rpow_absNorm_of_isArithGenuineCuspRealizable714 below · cited by 1 · depth 20 - Absolute summability of Whittaker coefficients on GL₂
AutomorphicForm.summable_norm_whittakerCoefficient_of_isKfSmooth_of_contDiff_mixedSpace12 below · cited by 4 · depth 20 - Whittaker transformation laws and torus ODE over ℚ
AutomorphicForm.whittakerCoefficient_archRealLiftAt_mul_laws_and_torus_ode_of_archCasimirAt_eq_smul_rat10 below · cited by 2 · depth 20 - Vanishing of the torus Whittaker function on the wrong sheet
AutomorphicForm.whittakerCoefficient_detOneTorus_eq_zero_of_iterate_lower_eq_zero6 below · cited by 4 · depth 20 - Non-vanishing at s=1 of twisted GL₂ Euler products over ℚ
AutomorphicForm.apply_one_ne_zero_of_differentiable_of_hasProd_eulerProduct_twist_of_norm_eq_one_rat677 below · cited by 1 · depth 21 - Casimir at a real place: translation and convolution invariance
AutomorphicForm.archCasimirAt_rightTranslate_and_rightConv_of_continuous_archDerivAt11 below · cited by 3 · depth 21 - J-rigid weight-one cut vector witnesses archimedean occurrence in the class
AutomorphicForm.archOccursInClassOf_J_rigid_of_mem_isCuspConstituent_of_hasArchCharacterAt_one358 below · cited by 1 · depth 21 - Square of the J-reflected lowering operator in weight one
AutomorphicForm.archReflectLower_archReflectLower_eq_smul_of_hasArchCharacterAt_one_of_archCasimirAt_eq_smul1 below · cited by 1 · depth 21 - Inert unit fundamental lemma for twisted GL₂
AutomorphicForm.areMatchingLocal_indicator_semiLocalIntegralSet_of_ramificationIdx_eq_one_of_inert_of_prime71 below · cited by 1 · depth 21 - The constant term intertwines right translation
AutomorphicForm.constantTerm_rightTranslate0 below · cited by 1 · depth 21 - Continuity of Borel-induced sections from the maximal compact
AutomorphicForm.continuousOn_of_isInducedSection_of_continuousOn_maximalCompact2 below · cited by 6 · depth 21 - Continuity of the big-cell Bruhat sum for Re s > 1/2
AutomorphicForm.continuous_bruhatTransversal_tsum_of_re_gt_half0 below · cited by 11 · depth 21 - Continuity and norm bound for GL₂ Whittaker coefficients
AutomorphicForm.continuous_whittakerCoefficient_and_exists_norm_le_mul_ideleNorm_det_rpow_of_isCuspAutomorphicFnAt_of_rightConv_eq75 below · cited by 1 · depth 21 - Vanishing on a set covering modulo GL₂(F) and the centre
AutomorphicForm.eq_zero_of_isLsXiFunction_of_coversModCentre_of_forall_mem_eq_zero0 below · cited by 1 · depth 21 - Isotypic cusp forms fixed by SL₂(Kᵥ) vanish
AutomorphicForm.eq_zero_of_mem_isotypicCuspSubmodule_of_forall_det_eq_one_invariant39 below · cited by 1 · depth 21 - Moderate growth across the centre of a continued Eisenstein family
AutomorphicForm.exists_analyticOnNhd_sub_mul_bruhatEisenstein_norm_le_archHeight_pow_of_ne_of_isArchKFinite_family163 below · cited by 1 · depth 21 - Archimedean transfer of twisted orbital integrals on GL₂
AutomorphicForm.exists_contDiff_hasCompactSupport_forall_isTwistedOrbitalIntegralOn_conjAe_imp_eq32 below · cited by 1 · depth 21 - Splitting an adelic GL₂ element along the good places
AutomorphicForm.exists_eq_mul_mem_levelOne_inf_finiteAdelicGL2Subgroup_commute_placeEmbed_of_forall_mem_localIntegralSet0 below · cited by 1 · depth 21 - Finiteness of double cosets meeting an orbital integrand's support
AutomorphicForm.exists_finset_forall_eq_and_forall_exists_of_isRegularSemisimple0 below · cited by 4 · depth 21 - Finiteness of twisted double cosets meeting the support
AutomorphicForm.exists_finset_forall_eq_and_forall_exists_of_isRegularSemisimple_normString1 below · cited by 2 · depth 21 - Finite convolution combination acting as identity on a cuspidal constituent
AutomorphicForm.exists_finset_sum_convOp_eq_self_of_isCuspConstituent166 below · cited by 1 · depth 21 - Nonzero twisted cut trace as a sum over fibre-constant cusp classes
AutomorphicForm.exists_finset_twistedCutTrace_eq_sum_twistedCutTrace_of_isFundamentalDomain_of_prime30 below · cited by 1 · depth 21 - Entire K-finite induced families are combinations of flat families
AutomorphicForm.exists_flat_isInducedSection_sum_eq_of_differentiable_family8 below · cited by 11 · depth 21 - Finite expansion of a K-finite smooth function on K
AutomorphicForm.exists_forall_rightTranslate_eq_sum_mul_of_isArchKFinite_of_isKfSmooth2 below · cited by 6 · depth 21 - Window-to-window L² domination between centre-cut Siegel sets
AutomorphicForm.exists_forall_setLIntegral_iUnion_centreCutSiegelSet_le_mul_of_coversModCentre_of_forall_ncard_le14 below · cited by 1 · depth 21 - Archimedean base-change transfer from a ramified real place
AutomorphicForm.exists_isArchTestFactor_forall_exists_isTwistedOrbitalIntegralOn_of_forall_exists_contDiff_conjAe11 below · cited by 1 · depth 21 - Occurring weight-one type lies in one cuspidal constituent
AutomorphicForm.exists_isCuspConstituent_mem_isotypicCuspSubmodule_archCutSubmodule_hasArchCharacterAt_one_of_archOccursInClassOf333 below · cited by 1 · depth 21 - Cuspidal realisations of Theta embed GL₂(ℚ_q)-equivariantly into one irreducible representation
AutomorphicForm.exists_isIrreducibleGLRep_linearMap_span_translate_realization_of_coversModCentre340 below · cited by 1 · depth 21 - Approximate identity of level-spherical flat test functions at compact level
AutomorphicForm.exists_isLevelSphericalOfType_flat_tendsto_rightConv_of_finiteDimensional_of_isCompact21 below · cited by 3 · depth 21 - Elliptic classes with degree-divisible determinant valuation are norms
AutomorphicForm.exists_isNormOf_of_not_isSquare_discr_of_finrank_dvd_of_ramificationIdx_eq_one17 below · cited by 1 · depth 21 - Convolution preserves semi-local factorisation at S
AutomorphicForm.exists_isSemiLocalFactorization_integral_mul_comp_inv_mul4 below · cited by 1 · depth 21 - Nonzero right convolution against a unit-factorizable test function
AutomorphicForm.exists_isUnitFactorizableAt_rightConv_ne_zero2 below · cited by 1 · depth 21 - Genuine cuspidal realizability of a Hecke eigenfunction over ℚ
AutomorphicForm.exists_level_isArithGenuineCuspRealizable_of_continuous_cuspidal_heckeEigen_rat122 below · cited by 1 · depth 21 - A J-rigid vector in weight-one Casimir eigenspaces
AutomorphicForm.exists_ne_zero_apply_mul_archRealGLAt_J_eq_mul_lower_of_finiteDimensional_of_forall_mem4 below · cited by 1 · depth 21 - Existence of a nonzero level-one invariant vector at v
AutomorphicForm.exists_ne_zero_forall_mem_localLevelOne_smul_eq_of_smooth_of_det_one_invariant_eq_zero0 below · cited by 1 · depth 21 - Equicontinuity of right convolutions on compact sets
AutomorphicForm.exists_nhds_one_forall_norm_rightConv_mul_sub_rightConv_le_mul_eLpNorm_of_isLsXiFunction_of_isFundamentalDomain12 below · cited by 1 · depth 21 - Rapid decay of smoothed cusp forms in the cusp
AutomorphicForm.exists_norm_rightConv_mul_le_mul_inv_archHeight_pow_of_lt_localHeight_of_isCuspAutomorphicFnAt_of_coversModCentre68 below · cited by 3 · depth 21 - Coordinatewise torus decay of Whittaker coefficients under a Casimir trichotomy
AutomorphicForm.exists_norm_whittakerCoefficient_diagOne_le_ideleNorm_rpow_of_pure_of_casimir_trichotomy20 below · cited by 2 · depth 21 - Positive lower bound for (σ-tfrac12)M(σ)φ_σ(1)
AutomorphicForm.exists_pos_eventually_le_re_sub_one_half_mul_weylIntertwiningIntegral_one_of_nonneg_of_isArchKFinite_family18 below · cited by 1 · depth 21 - Twisted diagonalisation with prescribed norms at a regular split element
AutomorphicForm.exists_twistedConj_eq_diagonal_and_norm_eq_of_conj_normString_eq_diagonal_of_ne0 below · cited by 1 · depth 21 - Comparison of twisted elliptic–central and kernel folds
AutomorphicForm.exists_twistedEllipticCentralFold_eq_mul_sum_kernelCentralEllipticFold901 below · cited by 1 · depth 21 - Continuation and decay of Bruhat–Eisenstein Whittaker coefficients
AutomorphicForm.exists_whittakerCoefficient_bruhatEisenstein_continuation_summable_norm_tsum_le_rpow_neg_of_isArchKFinite_family87 below · cited by 1 · depth 21 - Spectral comparison of cut traces in prime-degree Galois extensions
AutomorphicForm.fibreSum_twistedCutTrace_eq_const_mul_fibreSum_cutTrace_of_centralElliptic_of_prime3,001 below · cited by 1 · depth 21 - Formal base change satisfies the base-change relation
AutomorphicForm.isBaseChangeOf_formalBaseChange0 below · cited by 0 · depth 21 - Adelic height powers form a Borel-induced flat section
AutomorphicForm.isInducedSection_adelicHeight_cpow2 below · cited by 41 · depth 21 - Weyl intertwining integral reflects the inducing character pair
AutomorphicForm.isInducedSection_etaFst_etaSnd_neg_weylIntertwiningIntegral1 below · cited by 10 · depth 21 - Right convolution preserves isotypic cusp forms and archimedean cuts
AutomorphicForm.isIsotypicCuspFormAt_rightConv_of_isUnitFactorizableAt_of_forall_isHeckeCosetEigenfunctionAt73 below · cited by 2 · depth 21 - Split regular orbital integral as a coset sum
AutomorphicForm.isOrbitalIntegralOn_localHaar_mul_eq_finsum_indicator_of_heckeAlgebra_of_diagonal3 below · cited by 1 · depth 21 - Finite double-coset sum realises a local orbital integral
AutomorphicForm.isOrbitalIntegralOn_localHaar_sum_div_of_forall_eq_of_forall_exists0 below · cited by 2 · depth 21 - Twisted orbital integral as a finite double-coset sum
AutomorphicForm.isTwistedOrbitalIntegralOn_semiLocalHaar_sum_div_of_forall_eq_of_forall_exists0 below · cited by 2 · depth 21 - Vanishing of the isotypic cusp space when v∣ N
AutomorphicForm.isotypicCuspSubmodule_productionPinsOf_principal_eq_bot_of_dvd1 below · cited by 4 · depth 21 - Cauchy–Schwarz bound for right convolution on GL₂(A_K)
AutomorphicForm.norm_rightConv_le_eLpNorm_mul_eLpNorm_restrict_image_mul0 below · cited by 2 · depth 21 - Elliptic orbital integral of a spherical Hecke function via its Satake shadow
AutomorphicForm.orbitalIntegral_eq_shadow_of_irreducible_charpoly33 below · cited by 1 · depth 21 - Level-zero global additive characters of A_ℚ are locally ψᵥ^{± 1}
AutomorphicForm.psiLoc_eq_psiLocal_or_eq_inv_of_isGlobalAddChar_of_addCharLevel_eq_zero19 below · cited by 8 · depth 21 - Twisted orbital sum at an unramified place
AutomorphicForm.sum_relIndex_mul_twistedConj_diagonal_eq_zpow_absNorm_mul_finsum_of_ramificationIdx_eq_one1 below · cited by 1 · depth 21 - Summability of Whittaker coefficients of Bruhat Eisenstein series for Re s>1
AutomorphicForm.summable_whittakerCoefficient_bruhatEisenstein_of_one_lt_re_of_unitary25 below · cited by 1 · depth 21 - Leading term at s=1/2 of the Weyl intertwining integral is g-independent
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_sub_apply_one_nhds_zero_of_isArchKFinite_family93 below · cited by 1 · depth 21 - Twisted orbital integral at an inert place via Satake shadow
AutomorphicForm.twistedOrbitalIntegral_eq_shadow_of_irreducible_charpoly32 below · cited by 1 · depth 21 - Convergence of the Weyl intertwining integral for Re s > 1/2
AutomorphicForm.weylIntertwiningIntegrand_integrable_of_re_gt_half0 below · cited by 27 · depth 21 - Whittaker's equation for torus Whittaker coefficients at a real place
AutomorphicForm.whittakerCoefficient_diagOne_satisfies_whittaker_ode_of_archCasimirAt_eq_smul_of_hasArchCharacterAt5 below · cited by 4 · depth 21 - Whittaker coefficients of unipotent right translates at torus points
AutomorphicForm.whittakerCoefficient_finset_sum_mul_unipotentGL2_diagOne_mul1 below · cited by 3 · depth 21 - Deep congruence elements preserve U₁(N)-invariant functions on GL₂(A_K)
AutomorphicForm.apply_mul_eq_of_forall_mem_levelOne_of_valued_sub_one_le0 below · cited by 1 · depth 22 - Invariance of a translated level-one function under deep congruence elements
AutomorphicForm.apply_mul_mul_eq_of_forall_mem_levelOne_of_valued_sub_one_le_of_valued_apply_le0 below · cited by 1 · depth 22 - Casimir at a real place commutes with right convolution
AutomorphicForm.archCasimirAt_rightConv_of_isFactorizableTestFn_of_continuous_archDerivAt8 below · cited by 1 · depth 22 - Casimir at a real place commutes with right translation by GL₂(ℝ)
AutomorphicForm.archCasimirAt_rightTranslate_archRealGLAt0 below · cited by 3 · depth 22 - Casimir at a real place commutes with translations at other places
AutomorphicForm.archCasimirAt_rightTranslate_rowIsometryInclAt_of_ne0 below · cited by 3 · depth 22 - Invariance of the archimedean height under central scalars
AutomorphicForm.archHeight_glArch_centralScalar_mul0 below · cited by 2 · depth 22 - Big-cell values of a flat K-finite family as pure tensors
AutomorphicForm.bigCell_eq_sum_pureTensor_of_flat_family6 below · cited by 5 · depth 22 - Continuity of a family from its Iwasawa factorisation
AutomorphicForm.continuousOn_of_forall_apply_borel_mul_eq_of_continuousOn2 below · cited by 4 · depth 22 - Adjointness of double-coset translate sums for the weighted pairing
AutomorphicForm.cosetSum_adjoint_weightedPairing_of_isLsXiFunction10 below · cited by 6 · depth 22 - Analytic non-constant part of the Bruhat Eisenstein family
AutomorphicForm.exists_analyticOnNhd_bruhatEisenstein_sub_constantTerm_norm_le_rpow_neg_of_isArchKFinite_family_of_unitary116 below · cited by 3 · depth 22 - Regularised Weyl intertwining integral, distinct unitary characters
AutomorphicForm.exists_analyticOnNhd_sub_mul_weylIntertwiningIntegral_isInducedSection_of_ne_of_isArchKFinite_family109 below · cited by 1 · depth 22 - Non-vanishing of an induced section on the big cell
AutomorphicForm.exists_apply_weylInv_mul_unipotentGL2_ne_zero_of_isInducedSection_of_isKfSmooth0 below · cited by 1 · depth 22 - Per-place finite translate spans yield a bi-finite archimedean type
AutomorphicForm.exists_archTypeFamily_isArchFactorBiFinite_of_finiteDimensional_span_range0 below · cited by 1 · depth 22 - Adelic matching of orbital integrals in prime-degree base change
AutomorphicForm.exists_areMatchingOn_adeleRing_of_areMatchingAt_of_prime27 below · cited by 2 · depth 22 - Smooth compactly supported convolution over GL₂ of a normed field
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_isUnit_det_forall_eq_integral_of_contDiff0 below · cited by 1 · depth 22 - Atom-free trace identity with geometric remainder for GL₂
AutomorphicForm.exists_continuous_forall_not_isEisenstein_noAtomicMass_geometricRemainder1,284 below · cited by 2 · depth 22 - Twisted GL₂ trace identity with atom-free remainder functional
AutomorphicForm.exists_continuous_forall_not_isEisenstein_noAtomicMass_twistedGeometricRemainder_unram1,751 below · cited by 2 · depth 22 - Continuous part of a finite family of K-types
AutomorphicForm.exists_continuous_forall_typeSubmodule_le_iSup_and_range_eq_span_translates3 below · cited by 5 · depth 22 - Continuous Iwasawa decomposition of w⁻¹n(x) over the adeles
AutomorphicForm.exists_continuous_iwasawa_weyl_unipotent2 below · cited by 1 · depth 22 - Unitarity of the twisted Hecke table at almost all places
AutomorphicForm.exists_finset_forall_conj_mul_a_eq_div_of_isArithGenuineCuspRealizable_of_norm_twist_b_eq_one24 below · cited by 1 · depth 22 - Regularised partial Rankin–Selberg product over ℚ
AutomorphicForm.exists_finset_neg_analyticAt_ofReal_hasProd_rsEulerPoly_self_div_sub_one_rat663 below · cited by 1 · depth 22 - Uniform Haar bound for unipotent sweeps over a centre-cut Siegel set
AutomorphicForm.exists_forall_adelicGLHaar_image2_unipotentGL2_mul_mul_le_of_isCompact0 below · cited by 1 · depth 22 - Uniform square-integrability of right translates on a determinant slab
AutomorphicForm.exists_forall_memLp_two_comp_mul_right_restrict_and_eLpNorm_le_of_isFundamentalDomain12 below · cited by 3 · depth 22 - Split-or-inert dichotomy for L⊗_K Kᵥ in prime degree
AutomorphicForm.exists_idempotent_orbit_or_isField_tensor_adicCompletion0 below · cited by 2 · depth 22 - Averaging a right convolution over a compact level
AutomorphicForm.exists_integral_rightConv_eq_rightConv4 below · cited by 3 · depth 22 - Level average of a finite-adelic translate as Hecke coset sum
AutomorphicForm.exists_integral_rightTranslate_eq_inv_card_mul_sum1 below · cited by 3 · depth 22 - Conjugation transport of couplings and (twisted) orbital integrals
AutomorphicForm.exists_isHaarMeasure_coupled_isOrbitalIntegralOn_conj_and_isTwistedOrbitalIntegralOn_sigmaConj1 below · cited by 1 · depth 22 - Adelic GL₂ induced sections with prescribed K-type and support
AutomorphicForm.exists_isInducedSection_one_etaSnd_eq_on_maximalCompact_of_equivariant10 below · cited by 1 · depth 22 - Cuspidal constituents of GL₂/ℚ are isotypic at q
AutomorphicForm.exists_isIrreducibleGLRep_injective_linearMap_finsupp_of_isCuspConstituent165 below · cited by 1 · depth 22 - Inert case: elliptic γ is a σ-twisted norm
AutomorphicForm.exists_isNormOf_of_isField_tensor_adicCompletion_of_not_isSquare_discr_of_finrank_dvd14 below · cited by 1 · depth 22 - Local finite type on K for continuous families on adelic GL₂
AutomorphicForm.exists_isOpen_forall_exists_apply_eq_sum_of_isArchKFinite_of_continuous0 below · cited by 1 · depth 22 - Split transfer of twisted orbital integrals for GL₂
AutomorphicForm.exists_isTwistedSectionFnOn_integral_eq_fibreIntegral_of_isNormConjugator_one_of_mulEquiv_prod0 below · cited by 1 · depth 22 - Hecke and central translates of a factorizable test function
AutomorphicForm.exists_isUnitFactorization_insert_and_cutTrace_eq_pow_mul_cutTrace1 below · cited by 3 · depth 22 - A compensating unitary idele class character, with local triviality
AutomorphicForm.exists_isUnitaryChar_mul_conj_mul_eq_ideleNorm_rpow_of_admitsModulus7 below · cited by 1 · depth 22 - Existence of an archimedean type projector on GL₂(A_F)
AutomorphicForm.exists_linearMap_archCutProjector16 below · cited by 3 · depth 22 - Linear dependence of two torus Whittaker functions at a real place
AutomorphicForm.exists_ne_zero_forall_linearCombination_whittakerCoefficient_diagOne_eq_zero_of_archCasimirAt_eq_smul12 below · cited by 2 · depth 22 - Coordinatewise torus decay of Whittaker coefficients under Casimir trichotomy
AutomorphicForm.exists_norm_whittakerCoefficient_diagOne_le_ideleNorm_rpow_of_pure_of_casimir_trichotomy_of_finite_span20 below · cited by 1 · depth 22 - Whittaker decay at a complex place for an SU(2)-string
AutomorphicForm.exists_norm_whittakerCoefficient_diagOne_le_min_norm_rpow_of_isComplex_of_su2String18 below · cited by 1 · depth 22 - Adelic Weyl intertwining integral has a simple pole at σ=1/2
AutomorphicForm.exists_pos_eventually_le_sub_one_half_mul_setIntegral_adelicHeight_weyl_unipotent_rpow10 below · cited by 1 · depth 22 - Haar measure on GL₂(A_L^f) factors through semi-local components
AutomorphicForm.exists_pos_setIntegral_prod_semiLocalComponent_eq_mul_prod_integral0 below · cited by 1 · depth 22 - Tensor splitting of a right Kᵢ-finite function
AutomorphicForm.exists_sum_prod_mul_of_rightTranslatesSpanFinite0 below · cited by 2 · depth 22 - Hecke word shifts and scalar law for twisted cut traces
AutomorphicForm.exists_twistedCutTrace_heckeWordShift_eq_pow_mul_pow_mul2 below · cited by 3 · depth 22 - Twisted elliptic-central fold equals base-changed central-elliptic kernel
AutomorphicForm.exists_twistedEllipticCentralFold_eq_mul_sum_kernelCentralEllipticFold_of_areMatchingOn_of_isNormClass896 below · cited by 1 · depth 22 - Unipotent surgery concentrating a Whittaker coefficient on a small ball
AutomorphicForm.exists_unipotent_surgery_whittakerCoefficient_diagOne_mul_eq_sum_mul_ball9 below · cited by 1 · depth 22 - Continuation of Whittaker coefficients to Re s>0
AutomorphicForm.exists_whittakerCoefficient_diagOne_continuation_of_flat_family70 below · cited by 1 · depth 22 - Fibre-sum spectral comparison for twisted GL₂ at prime degree
AutomorphicForm.fibreSum_twistedCutTrace_eq_const_mul_fibreSum_cutTrace_of_docks_ed23,000 below · cited by 1 · depth 22 - Bi-finiteness yields finite-dimensional archimedean translate spans
AutomorphicForm.finiteDimensional_span_range_of_isArchFactorBiFinite0 below · cited by 1 · depth 22 - Whittaker's equation from the Casimir eigenrelation on GL₂(ℝ)
AutomorphicForm.gl2Real_whittaker_ode_of_casimir_of_unipotent_covariant_of_weight0 below · cited by 1 · depth 22 - Right translation by diag(1,-1) at a real place
AutomorphicForm.hasArchCharacterAt_neg_and_archCasimirAt_comp_mul_diag_one_neg_one0 below · cited by 1 · depth 22 - Integrability of a Bruhat majorant on translated centre-cut Siegel sets
AutomorphicForm.integrableOn_norm_sq_mul_bruhatMajorant_mul_ideleNorm_rpow_inter_centreCutSiegelSet8 below · cited by 1 · depth 22 - Integrability of the folded central–elliptic adelic GL₂ kernel
AutomorphicForm.integrableOn_setIntegral_mul_centralElliptic_adelicKernel_of_isFundamentalDomain_slab23 below · cited by 4 · depth 22 - Integrability of the Rankin–Selberg unfolding kernel on StimesGL₂(A)
AutomorphicForm.integrable_mul_apply_mul_conj_mul_ideleNorm_det_rpow_prod_restrict_of_memLp15 below · cited by 1 · depth 22 - Casimir at a real place of a right convolution
AutomorphicForm.isArchSmoothAt_rightConv_and_exists_archCasimirAt_rightConv_eq_of_isArchBiFinite14 below · cited by 1 · depth 22 - Smoothness and Casimir of a right convolution at a real place
AutomorphicForm.isArchSmoothAt_rightConv_and_exists_archCasimirAt_rightConv_eq_of_isArchBiFinite_ofChar9 below · cited by 1 · depth 22 - Left translates preserve factorizable bi-finite test functions
AutomorphicForm.isFactorizableTestFn_comp_inv_mul1 below · cited by 3 · depth 22 - Local–global principle for elliptic norm classes in GL₂
AutomorphicForm.isNormClass_mk_of_mem_ellipticCell_of_forall_isNormOf125 below · cited by 2 · depth 22 - Every class is a norm with trivial conjugator
AutomorphicForm.isNormConjugator_one_of_idempotent_orbit0 below · cited by 1 · depth 22 - Isotypic cusp forms: slab fundamental domain dominates Siegel windows
AutomorphicForm.isotypicCuspSubmodule_inf_archCutSubmodule_principalLevel_le_of_isFundamentalDomain_of_pos338 below · cited by 7 · depth 22 - Vanishing of level-one isotypic cusp spaces at primes dividing the level
AutomorphicForm.isotypicCuspSubmodule_productionPinsOf_levelOne_eq_bot_of_dvd1 below · cited by 1 · depth 22 - Finite-dimensional Hecke-stable cusp space lies in isotypic cut subspaces
AutomorphicForm.le_iSup_isotypicCuspSubmodule_inf_archCutSubmodule_of_finiteDimensional_of_forall_heckeCosetSum_mem16 below · cited by 2 · depth 22 - Iwasawa integration over torus shells on GL₂(Kᵥ)
AutomorphicForm.lintegral_mul_density_eq_tsum_torusShells_localGL23 below · cited by 7 · depth 22 - Polynomial decay of adelic Weyl–unipotent integrals, unitary twists
AutomorphicForm.norm_integral_weyl_unipotent_mul_addChar_le_polyDecay_of_unitary21 below · cited by 1 · depth 22 - Uniform torus bounds on Whittaker coefficients pass to archimedean translates
AutomorphicForm.norm_whittakerCoefficient_translate_diagOne_mul_le_of_glFin_eq_one8 below · cited by 3 · depth 22 - Petersson pairing of a right convolution, by Fubini
AutomorphicForm.peterssonIntegral_rightConv_eq_integral_mul_peterssonIntegral_translate0 below · cited by 1 · depth 22 - Zeroth shells outside S and the S-part torus measure
AutomorphicForm.setLIntegral_rationalCentreUnipotentQuotientMeasure_shellZeroOutside_eq_mul_lintegral_sPartMeasure8 below · cited by 1 · depth 22 - Summability of twisted cut traces over cuspidal classes
AutomorphicForm.summable_norm_twistedCutTrace_of_isFactorizableTestFn_of_isFundamentalDomain_slab94 below · cited by 3 · depth 22 - Intertwining residue at s=1/2 agrees on the maximal compact
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_sub_apply_one_nhds_zero_of_mem_maximalCompact92 below · cited by 1 · depth 22 - Twisted centralizer of a norm of regular semisimple γ
AutomorphicForm.twistedCentralizer_eq_map_centralizer_of_isNormConjugator_one0 below · cited by 1 · depth 22 - Uniform discreteness of GL₂(F) in GL₂(A_F)
AutomorphicForm.adelicKernelLocalFiniteness0 below · cited by 50 · depth 23 - Type membership transports along an equivariant operator
AutomorphicForm.apply_mem_iSup_typeSubmodule_of_isRightEquivariant_of_injective1 below · cited by 7 · depth 23 - Vanishing of the six complex flow derivatives forces SL₂(ℂ)-invariance
AutomorphicForm.apply_mul_archComplexGLAt_eq_of_forall_archDerivAtComplex_eq_zero0 below · cited by 2 · depth 23 - Sum of the two Casimir operators at a complex place
AutomorphicForm.archCasimirAtComplex_add_archCasimirBarAtComplex_eq_of_isArchSmoothAtComplex1 below · cited by 2 · depth 23 - Casimir operators at a complex place commute with translation and convolution
AutomorphicForm.archCasimirAtComplex_rightTranslate_and_rightConv_of_continuous_archDerivAtComplex12 below · cited by 2 · depth 23 - Symmetric 𝔭-identity at a complex place, lowest-weight form
AutomorphicForm.archDelAt_E_archDelAt_Fm_add_archDelBarAt_Fm_archDelBarAt_E_eq_of_archDerivAtComplex_iH_eq_smul1 below · cited by 2 · depth 23 - Symmetric 𝔭-identity at a complex place on an iH-eigenvector
AutomorphicForm.archDelAt_Fm_archDelAt_E_add_archDelBarAt_E_archDelBarAt_Fm_eq_of_archDerivAtComplex_iH_eq_smul1 below · cited by 2 · depth 23 - Compact-direction derivatives vanish for trivial character at a complex place
AutomorphicForm.archDerivAtComplex_iH_eq_zero_and_Fm_eq_E_and_iFm_eq_neg_iE_of_hasArchCharacterAtZero_one0 below · cited by 2 · depth 23 - Compact Casimir acts by -(n²+2n) on highest-weight vectors
AutomorphicForm.archKCasimirAtComplex_eq_smul_of_archDerivAtComplex_iH_eq_smul_of_compactRaise_eq_zero1 below · cited by 1 · depth 23 - Admissibility of the canonical truncation datum for GL₂
AutomorphicForm.canonicalTruncationData_isTruncationDatum17 below · cited by 109 · depth 23 - Spectral side of the twisted trace formula along Hecke words
AutomorphicForm.exists_atomic_forall_exists_integral_lambdaT_twistedAdelicKernel_eq_twistedCutTrace_add_symm_unram1,750 below · cited by 1 · depth 23 - Truncated GL₂ kernel integral along Hecke words: affine asymptotics
AutomorphicForm.exists_atomic_forall_tendsto_setIntegral_lambdaT_adelicKernel_sub_mul1,282 below · cited by 1 · depth 23 - Hecke word comparison of twisted and untwisted cut traces
AutomorphicForm.exists_atoms_forall_exists_noAtomicMass_heckeWordSum_twistedCutTrace_sub_finrank_mul_const_mul_heckeWordSum_cutTrace_eq2,972 below · cited by 1 · depth 23 - Archimedean induced section at (1,ν) with prescribed K_∞-type
AutomorphicForm.exists_continuous_isArchKFinite_eq_of_borel_arch_of_equivariant5 below · cited by 1 · depth 23 - Continuous matrix representations separate points of archimedean row-isometry groups
AutomorphicForm.exists_continuous_monoidHom_matrix_apply_ne_one_of_ne_one0 below · cited by 1 · depth 23 - Coordinate matrices for an SU(2)-string at a complex place
AutomorphicForm.exists_coordMatrix_rightTranslate_rot_of_linearIndependent_of_span_stable0 below · cited by 1 · depth 23 - Place-by-place coupling of adelic and twisted orbital measures
AutomorphicForm.exists_coupled_smul_and_eq_mul_prod_of_coupled_adeleRing1 below · cited by 2 · depth 23 - Formal base change of an Eisenstein Hecke table is Eisenstein
AutomorphicForm.exists_eisensteinTableOf_eq_formalBaseChange_eisensteinTableOf6 below · cited by 1 · depth 23 - Factorizable test functions as finite sums of convolutions
AutomorphicForm.exists_eq_sum_rightConv_of_isFactorizableTestFn5 below · cited by 7 · depth 23 - Prime-degree extension inert at v when L⊗_K Kᵥ is a field
AutomorphicForm.exists_extension_algEquiv_adicCompletion_of_isField_tensor0 below · cited by 1 · depth 23 - Independent tensor splitting of the finite Whittaker factor
AutomorphicForm.exists_finWhittaker_eq_sum_prod_mul_linearIndependent_levelOne_invariant_of_isIsotypicCuspFormAt_of_localSpaceAt15 below · cited by 1 · depth 23 - Base change for GL₂: elliptic–central class sums compared
AutomorphicForm.exists_finsum_sigmaCentralizerDomain_eq_mul_sum_finsum_centralizerDomain_of_areMatchingOn_of_isNormClass867 below · cited by 1 · depth 23 - Rank-one torus Whittaker functions of a complex-place SU(2)-string
AutomorphicForm.exists_forall_whittakerCoefficient_diagOne_eq_mul_of_isComplex_of_su2String14 below · cited by 1 · depth 23 - Haar measure on GL₂(L⊗_KA_K) factorises over places of K
AutomorphicForm.exists_integral_baseChange_eq_mul_integral_mul_prod_integral_semiLocalHaar_of_isHaarMeasure0 below · cited by 5 · depth 23 - Haar measure on adelic GL₂-centralizers factors over places
AutomorphicForm.exists_integral_centralizer_eq_mul_integral_mul_prod_integral_of_isHaarMeasure0 below · cited by 2 · depth 23 - Adelic Haar integral factors over the places of GL₂
AutomorphicForm.exists_integral_eq_mul_integral_mul_prod_integral_localHaar_of_isHaarMeasure0 below · cited by 6 · depth 23 - Haar measure on a twisted centralizer factorises over the places
AutomorphicForm.exists_integral_twistedCentralizer_eq_mul_integral_mul_prod_integral_of_isHaarMeasure0 below · cited by 5 · depth 23 - Non-vanishing pairing of a ν-covariant kernel with an arch-finite function
AutomorphicForm.exists_isArchKFinite_equivariant_integral_maximalCompactAtHaar_mul_ne_zero2 below · cited by 2 · depth 23 - Non-negative K_∞-finite function pairing non-trivially with β
AutomorphicForm.exists_isArchKFinite_invariant_nonneg_integral_maximalCompactAtHaar_mul_ne_zero2 below · cited by 1 · depth 23 - A compact carrier for Satake boxes and their formal base change
AutomorphicForm.exists_isCompact_carrier_box_union_formalBaseChange0 below · cited by 1 · depth 23 - Approximate identity by factorizable test functions at principal level
AutomorphicForm.exists_isFactorizableTestFn_principalLevel_tendsto_rightConv0 below · cited by 2 · depth 23 - A K_f-smooth induced section with prescribed level and support
AutomorphicForm.exists_isKfSmooth_eq_prod_localChar_of_borel_fin_of_level5 below · cited by 1 · depth 23 - Adelic norms from local norms in prime-degree base change
AutomorphicForm.exists_isNormOf_adeleRing_of_forall_exists_isNormOf_of_prime8 below · cited by 2 · depth 23 - Factorisation of adelic orbital integrals of unit-factorizable functions
AutomorphicForm.exists_isOrbitalIntegralOn_adeleRing_eq_mul_prod_of_isUnitFactorization6 below · cited by 4 · depth 23 - Euler factorisation of a global twisted orbital integral
AutomorphicForm.exists_isTwistedOrbitalIntegralOn_baseChange_eq_mul_prod_of_isSemiLocalFactorization6 below · cited by 3 · depth 23 - Splitting an adelic maximal compact element at and away from S
AutomorphicForm.exists_mem_maximalCompactAt_mul_mem_maximalCompactAway_eq0 below · cited by 3 · depth 23 - Equivariant archimedean cut projectors transport right convolutions
AutomorphicForm.exists_rightConv_eq_of_archCutProjector14 below · cited by 1 · depth 23 - Rankin–Selberg package for Theta×̃Theta over ℚ
AutomorphicForm.exists_rs22GlobalIntegral_godementEisenstein_self_eq_add_div_and_mul_hasProd_rsEulerPoly_self_rat662 below · cited by 1 · depth 23 - mathfraksl₂-string basis for SU(2)-stable spaces at a complex place
AutomorphicForm.exists_su2Strings_of_finiteDimensional_of_isArchSmoothAtComplex3 below · cited by 1 · depth 23 - Uniform Iwasawa coordinates for isometric translates of g
AutomorphicForm.exists_uniform_iwasawa_mul_of_glFin_eq_one2 below · cited by 1 · depth 23 - Whittaker coefficients of a Bruhat–Eisenstein family: continuation and decay
AutomorphicForm.exists_whittakerCoefficient_bruhatEisenstein_continuation_summable_norm_tsum_le_rpow_neg_of_isArchKFinite_family_of_unitary87 below · cited by 1 · depth 23 - Finite-dimensionality of the principal-level isotypic cusp space on a window
AutomorphicForm.finiteDimensional_isotypicCuspSubmodule_principal_inf_archCutSubmodule337 below · cited by 1 · depth 23 - Fibre-sum vanishing from monomial identities at places of record
AutomorphicForm.forall_finset_fibreSum_sub_const_mul_fibreSum_add_eq_zero_of_forall_places_exists_noAtomicMass_wordSum_eq1 below · cited by 1 · depth 23 - Formal base change is constant on Galois fibres
AutomorphicForm.formalBaseChange_a_b_eq_of_under_eq0 below · cited by 3 · depth 23 - Torus Whittaker system for an SU(2)-string on GL₂(ℂ)
AutomorphicForm.gl2Complex_whittaker_system_of_casimir_pair_of_unipotent_covariant_of_circleWeight_of_ktype0 below · cited by 2 · depth 23 - Vanishing of the three compact derivatives forces trivial character at a complex place
AutomorphicForm.hasArchCharacterAtZero_one_of_archDerivAtComplex_compact_eq_zero2 below · cited by 1 · depth 23 - Derivatives at a complex place pass through Whittaker coefficients
AutomorphicForm.hasDerivAt_whittakerCoefficient_archFlowComplex_of_continuous_archDerivAtComplex0 below · cited by 2 · depth 23 - Iwasawa shell expansion of the N-quotient integral on GL₂(Kᵥ)
AutomorphicForm.hasSum_integral_torusShells_of_integrable_withDensity_density_localGL24 below · cited by 7 · depth 23 - Galois invariance of the idelic norm of det on GL₂(A_E)
AutomorphicForm.ideleNorm_det_sigmaAdelicAct0 below · cited by 31 · depth 23 - Integrability of the elliptic kernel diagonal on a determinant slab
AutomorphicForm.integrableOn_adelicKernelEllipticPart_diag_of_isFundamentalDomain_slab17 below · cited by 2 · depth 23 - Left Casimir preserves test functions, types and level
AutomorphicForm.isFactorizableTestFn_leftCasimir_and_rightConv_mem_of_isArchBiFinite11 below · cited by 1 · depth 23 - Casimir of a test function: level and archimedean type
AutomorphicForm.isFactorizableTestFn_leftCasimir_and_rightConv_mem_of_isArchBiFinite_ofChar6 below · cited by 1 · depth 23 - Smoothing isotypic cusp forms into a Siegel-window isotypic space
AutomorphicForm.rightConv_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_isFundamentalDomain_of_pos77 below · cited by 2 · depth 23 - Right convolutions commute at one irreducible archimedean type
AutomorphicForm.rightConv_rightConv_comm_of_isLevelSphericalOfType_of_isCompact1 below · cited by 1 · depth 23 - Satake data constant on fibres over K, given word-shift
AutomorphicForm.satakeData_eq_of_under_eq_of_twistedCutTrace_ne_zero_of_heckeWordShift0 below · cited by 3 · depth 23 - Adjointness of partial_X,partial̄_X and the Casimir operators at a complex place
AutomorphicForm.setIntegral_archCasimirAtComplex_mul_conj_eq_and_archDelAt_adjoint_of_isFundamentalDomain19 below · cited by 2 · depth 23 - Skew-symmetry of complex-place flow derivatives against the Petersson pairing
AutomorphicForm.setIntegral_archDerivAtComplex_mul_conj_add_eq_zero_of_isFundamentalDomain17 below · cited by 4 · depth 23 - Unfolding the central and elliptic terms of the adelic GL₂ kernel
AutomorphicForm.setIntegral_centralEllipticFold_eq_finsum_inv_card_mul_integral_setIntegral_centralizerDomain31 below · cited by 2 · depth 23 - Twisted elliptic–central fold as weighted twisted orbital integrals
AutomorphicForm.setIntegral_twistedCentralEllipticFold_eq_finsum_inv_card_mul_setIntegral_sigmaCentralizerDomain100 below · cited by 2 · depth 23 - Galois action permutes local factors and Hecke generators
AutomorphicForm.sigmaAdelicAct_localEmbed_range_and_heckeGen_of_asIdeal_eq_smul0 below · cited by 1 · depth 23 - Finite-dimensional mathfraksu(2)-calculus and circle-weight decomposition at a complex place
AutomorphicForm.su2Derivs_stable_and_hasDerivAt_and_exists_sum_hasCircleWeightAt_of_finiteDimensional0 below · cited by 4 · depth 23 - Absolute summability of Siegel-pinned cut traces on GL₂
AutomorphicForm.summable_norm_cutTrace_of_isUnitFactorizableOfTypeAt_of_coversModCentre96 below · cited by 1 · depth 23 - Satake table of a principal-level cuspidal class lies in a box
AutomorphicForm.table_mem_box_of_mem_cuspClasses_siegel119 below · cited by 1 · depth 23 - Hecke tables of cuspidal slab classes lie in the box
AutomorphicForm.table_mem_box_of_mem_cuspClasses_slab18 below · cited by 1 · depth 23 - Flat families: intertwining integral residue independent of K-variable
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_sub_apply_one_nhds_zero_of_flat_family91 below · cited by 1 · depth 23 - Nonvanishing Haar mass of a norm band of idele classes
AutomorphicForm.toReal_measure_inter_setOf_ideleNorm_det_centralScalar_mem_Icc_ne_zero7 below · cited by 1 · depth 23 - From record-level identities to the fibre identity off S_L
AutomorphicForm.tsum_fibre_eq_const_mul_sum_tsum_fibre_formalBaseChange_of_forall_finset_record_of_subset0 below · cited by 1 · depth 23 - Integrability of the Weyl intertwining integrand over finite adeles
AutomorphicForm.weylIntertwiningIntegrand_finiteAdeleSlice_integrable_of_re_gt_half1 below · cited by 1 · depth 23 - Euler-product shape of Whittaker coefficients of a flat Eisenstein family
AutomorphicForm.whittakerCoefficient_bruhatEisenstein_diagOne_eq_cpowChar_mul_sum_eulerProduct_of_flat_family57 below · cited by 1 · depth 23 - Complex-place Casimirs commute with right convolution by test functions
AutomorphicForm.archCasimirAtComplex_rightConv_of_isFactorizableTestFn_of_continuous_archDerivAtComplex9 below · cited by 1 · depth 24 - Complex-place Casimir operators commute with right translation
AutomorphicForm.archCasimirAtComplex_rightTranslate_archComplexGLAt0 below · cited by 3 · depth 24 - Complex-place Casimir operators commute with translation at other places
AutomorphicForm.archCasimirAtComplex_rightTranslate_rowIsometryInclAt_of_ne0 below · cited by 3 · depth 24 - Commutation relations for the six flow derivatives at a complex place
AutomorphicForm.archDerivAtComplex_commutator_of_isArchSmoothAtComplex0 below · cited by 9 · depth 24 - Right convolution at a complex place: smoothing and integration by parts
AutomorphicForm.archDerivAtComplex_rightConv_eq_rightConv_deriv_of_isFactorizableTestFn3 below · cited by 10 · depth 24 - Hecke words and slot-family combinations are matching at S_K∪ T
AutomorphicForm.areMatchingAt_union_heckeWord_sum_slotFamilyCoeff_mul_of_areMatchingAt78 below · cited by 3 · depth 24 - Continuous kernel orthogonal to all arch. K-finite functions vanishes
AutomorphicForm.eq_zero_of_continuous_of_forall_isArchKFinite_integral_maximalCompactAtHaar_mul_eq_zero1 below · cited by 2 · depth 24 - Adelic matching for prime-degree cyclic base change of GL₂
AutomorphicForm.exists_areMatchingOn_and_central_adeleRing_of_areMatchingAt_of_prime715 below · cited by 1 · depth 24 - Asymptotic twisted spectral identity for GL₂, ramified places in S_L
AutomorphicForm.exists_atomic_forall_tendsto_integral_lambdaT_twistedAdelicKernel_sub_twistedCutTrace_sub_unram1,335 below · cited by 1 · depth 24 - Spectral side of the truncated centre-folded GL₂ trace formula
AutomorphicForm.exists_atomic_forall_tendsto_setIntegral_lambdaT_finsum_integral_centralScalar_sub_mul1,280 below · cited by 1 · depth 24 - Norm conjugator identifies centralizer with twisted centralizer
AutomorphicForm.exists_continuousMulEquiv_centralizer_twistedCentralizer_of_isNormConjugator0 below · cited by 5 · depth 24 - Twisted geometric remainder minus [L:K]λ times slot sum: cylinder-small functional
AutomorphicForm.exists_continuous_noAtomicMass_twistedGeometricRemainder_sub_finrank_mul_const_mul_sum_eq1,677 below · cited by 1 · depth 24 - Global section function yields local section data at every place
AutomorphicForm.exists_finset_forall_isSectionFnOn_indicator_localIntegralSet_of_isSectionFnOn_adeleRing0 below · cited by 2 · depth 24 - Semi-local twisted section data for a factorisable function
AutomorphicForm.exists_finset_forall_isTwistedSectionFnOn_indicator_semiLocalIntegralSet_of_isTwistedSectionFnOn_baseChange0 below · cited by 2 · depth 24 - Uniform level for a flat family of induced sections
AutomorphicForm.exists_forall_apply_mul_eq_of_mem_maximalCompactAway_of_flat_family3 below · cited by 1 · depth 24 - Haar measure on GL₂(Kᵥ) in Bruhat big-cell coordinates
AutomorphicForm.exists_haar_localGL2_eq_smul_map_lowerUnipotentGL2_mul_diagUnits2_mul_unipotentGL23 below · cited by 4 · depth 24 - Averaging a bottom-row valuation condition over the adelic maximal compact
AutomorphicForm.exists_lintegral_ite_bottomRow_maximalCompactHaar_eq_mul_lintegral_maximalCompactAtHaar_empty0 below · cited by 1 · depth 24 - Peeling off the component at one finite place
AutomorphicForm.exists_mem_maximalCompactAt_erase_mul_eq_of_mem_maximalCompactAt0 below · cited by 1 · depth 24 - Contragredient-type representative with unchanged right convolution
AutomorphicForm.exists_mem_span_rightTranslate_mem_archDualCutSubmodule_and_rightConv_eq6 below · cited by 1 · depth 24 - Integral norm-string witnesses at unramified places of prime degree
AutomorphicForm.exists_normString_eq_toTensorGL_of_mem_localIntegralSet_of_ramificationIdx_eq_one_of_prime7 below · cited by 2 · depth 24 - Rankin–Selberg test data over ℚ for a cusp-realizable eigensystem
AutomorphicForm.exists_rankinSelberg_testData_of_isArithGenuineCuspRealizable_rat554 below · cited by 1 · depth 24 - Continuation of Whittaker coefficients of a unitary flat Eisenstein family
AutomorphicForm.exists_whittakerCoefficient_diagOne_continuation_of_flat_family_of_unitary70 below · cited by 1 · depth 24 - Base-independent multiplier for torus Whittaker coefficients at a complex place
AutomorphicForm.exists_whittakerCoefficient_diagOne_eq_mul_whittakerCoefficient_splitTorusGL2Complex_of_hasCircleWeightAt4 below · cited by 1 · depth 24 - Finite-dimensionality of isotypic cusp spaces at principal level
AutomorphicForm.finiteDimensional_isotypicCuspSubmodule_principal_inf_archCutSubmodule_of_isFundamentalDomain336 below · cited by 3 · depth 24 - K-finiteness of vectors in the archimedean dual cut
AutomorphicForm.finiteDimensional_span_rightTranslate_of_mem_archDualCutSubmodule0 below · cited by 1 · depth 24 - Finiteness of fibres of (a,b)↦(p_f(a,b),b^f)
AutomorphicForm.finite_preimage_satakePow_pow0 below · cited by 1 · depth 24 - Finiteness of twisted elliptic and central classes meeting a compact support
AutomorphicForm.finite_sep_exists_twistedKernelSummand_ne_zero_of_hasCompactSupport11 below · cited by 3 · depth 24 - Central-norm twisted terms versus central terms, prime degree
AutomorphicForm.finsum_sigmaCentralizerDomain_centralNorm_eq_mul_sum_finsum_centralizerDomain_central_of_central_transfer310 below · cited by 2 · depth 24 - Cyclic base change: elliptic-norm twisted terms versus elliptic terms
AutomorphicForm.finsum_sigmaCentralizerDomain_ellipticNorm_eq_mul_sum_finsum_centralizerDomain_elliptic_of_areMatchingOn_of_eq_zero285 below · cited by 2 · depth 24 - Right translation by a maximal compact element preserves flat families
AutomorphicForm.flat_family_comp_mul_of_mem_adelicMaximalCompact0 below · cited by 3 · depth 24 - Central–elliptic geometric side of a truncated twisted trace formula
AutomorphicForm.forall_exists_integral_lambdaT_twistedAdelicKernel_eq_finsum_centralElliptic_add_and_norm_le_unram569 below · cited by 1 · depth 24 - Twisted orbital expansion of the elliptic and central kernel part
AutomorphicForm.hasSum_setIntegral_setIntegral_twistedOrbital_of_normClass_elliptic_or_central6 below · cited by 1 · depth 24 - Class-by-class expansion of a twisted GL₂ kernel integral
AutomorphicForm.hasSum_setIntegral_sigmaCentralizer_of_lintegral_tsum_enorm_lt_top0 below · cited by 2 · depth 24 - Per-word twisted spectral comparison from the remainder rows
AutomorphicForm.heckeWordSum_twistedCutTrace_sub_const_mul_heckeWordSum_cutTrace_add_atoms_eq_of_remainder_rows_of_comparison194 below · cited by 1 · depth 24 - Folding the centre out of the truncated adelic kernel
AutomorphicForm.integrableOn_and_setIntegral_mul_lambdaT_adelicKernel_centralScalar_mul_eq_lambdaT_finsum11 below · cited by 3 · depth 24 - Casimir action on a smoothing at a complex place
AutomorphicForm.isArchSmoothAtComplex_rightConv_and_exists_archCasimirAtComplex_rightConv_eq_of_isArchBiFinite15 below · cited by 1 · depth 24 - Finite sums of archimedean two-sided translates stay factorizable
AutomorphicForm.isFactorizableTestFn_sum_mul_comp_mul_mul0 below · cited by 1 · depth 24 - Norms in GL₂ are preserved by coefficient homomorphisms
AutomorphicForm.isNormOf_map_of_isNormOf0 below · cited by 3 · depth 24 - Product of local section functions is a global section function
AutomorphicForm.isSectionFnOn_adeleRing_indicator_prod_of_forall_isSectionFnOn0 below · cited by 2 · depth 24 - Products of local twisted section functions are global
AutomorphicForm.isTwistedSectionFnOn_baseChange_indicator_prod_of_forall_isTwistedSectionFnOn0 below · cited by 3 · depth 24 - Type pieces are stable under equivariant surjections of finite-dimensional stable subspaces
AutomorphicForm.le_iSup_typeSubmodule_of_surjective_of_le_iSup_typeSubmodule1 below · cited by 3 · depth 24 - Finiteness of the elliptic–central part of the twisted kernel
AutomorphicForm.lintegral_lintegral_tsum_enorm_twistedKernel_normClass_elliptic_or_central_lt_top93 below · cited by 3 · depth 24 - Hecke word evaluation on adelic induced sections
AutomorphicForm.rightConv_eq_prod_pow_mul_pow_mul_rightConv_of_isInducedSection_of_isUnitFactorization4 below · cited by 2 · depth 24 - Nonvanishing weighted L² self-pairing over a slab fundamental domain
AutomorphicForm.setIntegral_mul_conj_mul_ideleNorm_det_rpow_ne_zero_of_isLsXiFunction_of_isFundamentalDomain3 below · cited by 4 · depth 24 - Slot-family expansion of base-changed Hecke words over a finite set of places
AutomorphicForm.sum_slotFamilyCoeff_mul_prod_pow_mul_pow_eq_prod_eval_slotWord_div0 below · cited by 2 · depth 24 - Godement's lemma: absolute convergence of the Bruhat series for Re s>1/2
AutomorphicForm.summable_norm_godementSection_adelicWeyl_unipotentGL2_mul_of_mem_schwartzBruhat2_of_half_lt_re60 below · cited by 1 · depth 24 - Flat families: intertwining integral residue at 1/2 is K_∞-invariant
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_sub_nhds_zero_of_flat_family_of_mem_maximalCompactAt_empty79 below · cited by 1 · depth 24 - Leading term at s=1/2 of intertwining integral is Kᵥ-invariant
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_sub_nhds_zero_of_flat_family_of_mem_maximalCompactAt_singleton73 below · cited by 1 · depth 24 - Hypotheses of the GL₂(ℂ) Whittaker system for an SU(2)-string
AutomorphicForm.whittakerCoefficient_su2String_gl2Complex_whittaker_system_hypotheses5 below · cited by 1 · depth 24 - Finiteness of the Haar measure of a translated Siegel set in a determinant slab over ℚ
AutomorphicForm.adelicGLHaar_image_mul_right_integralWindowedSiegelSet_inter_slab_lt_top_rat3 below · cited by 1 · depth 25 - Matching of the Hecke word T_w^k z_w^j under prime-degree base change
AutomorphicForm.areMatchingLocal_heckeWord_sum_coeff_univWord_mul_heckeWord_of_ramificationIdx_eq_one_of_prime77 below · cited by 4 · depth 25 - Adelic matching of orbital integrals for prime-degree base change on GL₂
AutomorphicForm.areMatchingOn_and_central_adeleRing_of_areMatchingAt_of_prime_of_factorization713 below · cited by 4 · depth 25 - Residual block of the truncated GL₂ kernel: Eisenstein atoms
AutomorphicForm.exists_atomic_forall_integrableOn_and_tendsto_setIntegral_lambdaT_finsum_chiDet_mul_chiDet_inv26 below · cited by 1 · depth 25 - Atomic spectral data for the twisted truncated GL₂ trace
AutomorphicForm.exists_atomic_forall_tendsto_integral_lambdaT_twistedAdelicKernel_sub_twistedCutTrace_sub1,334 below · cited by 1 · depth 25 - Eisenstein block of the truncated centre-folded GL₂ kernel
AutomorphicForm.exists_atomic_forall_tendsto_setIntegral_lambdaT_finsum_sub_lambdaT_tsum_sub_lambdaT_finsum_chiDet_sub_mul1,277 below · cited by 1 · depth 25 - Bruhat decomposition of integral GL₂ at a finite place
AutomorphicForm.exists_borel_mul_gl2Weyl_mul_unipotentGL2_eq_of_integral0 below · cited by 1 · depth 25 - Comparison of parabolic intercepts along Hecke words, uniform λ
AutomorphicForm.exists_continuous_noAtomicMass_intercept_parabolic_sub_finrank_mul_const_mul_sum_intercept_parabolic_eq_uniform1,673 below · cited by 1 · depth 25 - Peeling one archimedean place off a maximal compact element
AutomorphicForm.exists_eq_mul_archSupportedAt_of_mem_maximalCompactAt_empty0 below · cited by 1 · depth 25 - Finitely many cuspidal constituents meet a principal-level eigensystem
AutomorphicForm.exists_finset_isCuspConstituent_le_iSup_of_cuspConstituentMeets_principal244 below · cited by 1 · depth 25 - Uniform bound on elliptic–central twisted kernel terms off the identity family
AutomorphicForm.exists_forall_encard_setOf_twistedKernelSummand_ne_zero_not_identityFamily_le23 below · cited by 1 · depth 25 - Coarse geometric expansion of the truncated GL₂ kernel integral
AutomorphicForm.exists_forall_le_setIntegral_lambdaT_adelicKernel_sub_centralElliptic_eq_setIntegral_parabolic94 below · cited by 3 · depth 25 - Coarse geometric expansion of the truncated twisted GL₂ kernel
AutomorphicForm.exists_forall_le_setIntegral_lambdaT_twistedAdelicKernel_sub_centralElliptic_eq_setIntegral_parabolic157 below · cited by 1 · depth 25 - Haar measure on GL₂(Kᵥ) in big Bruhat cell coordinates
AutomorphicForm.exists_haar_localGL2_eq_smul_map_unipotentGL2_mul_diagUnits2_mul_lowerUnipotentGL24 below · cited by 2 · depth 25 - Orbital integral at a central element of GL₂(A_K)
AutomorphicForm.exists_isHaarMeasure_and_isOrbitalIntegralOn_centralScalar_smul_adelicGLHaar1 below · cited by 1 · depth 25 - Approximate identity of unit-factorizable test functions at principal level
AutomorphicForm.exists_isUnitFactorizableAboveOfType_principalLevel_tendsto_rightConv_of_mem_archCutSubmodule1 below · cited by 1 · depth 25 - Orthonormal Hecke-adapted basis of the cut cuspidal space
AutomorphicForm.exists_orthonormal_isotypicCuspSubmodule_principalLevel_of_isFundamentalDomain_slab340 below · cited by 3 · depth 25 - Shaped unitary cusp vector over ℚ with factorised Whittaker function
AutomorphicForm.exists_unitaryShapedVector_whittakerFactorization_torusProfile_of_isArithGenuineCuspRealizable_rat544 below · cited by 1 · depth 25 - Admissibility of a cuspidal constituent at principal level
AutomorphicForm.finiteDimensional_inf_levelInvariantSubmodule_principal_inf_archCutSubmodule_of_isCuspConstituent162 below · cited by 7 · depth 25 - Elliptic twisted terms assemble into the base-change elliptic sum
AutomorphicForm.finsum_sigmaCentralizerDomain_ellipticNorm_eq_mul_sum_finsum_centralizerDomain_elliptic_of_forall_perClass163 below · cited by 1 · depth 25 - Type splitting of the truncated twisted kernel, with domain independence
AutomorphicForm.forall_exists_lambdaT_twistedAdelicKernel_eq_finsum_add_sub_indicator_constantTerm_add15 below · cited by 2 · depth 25 - Hyperbolic term affine in the truncation parameter, with bounded coefficients
AutomorphicForm.forall_exists_setIntegral_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_affine_bound415 below · cited by 1 · depth 25 - Affineness in R of the truncated twisted unipotent term
AutomorphicForm.forall_exists_setIntegral_finsum_unipotentCell_sub_indicator_constantTerm_eq_affine_unram257 below · cited by 3 · depth 25 - Cuspidal block of the truncated GL₂ spectral side
AutomorphicForm.forall_integrableOn_and_setIntegral_lambdaT_mul_tsum_convOp_mul_conj_eq_mul_tsum_cutTrace505 below · cited by 2 · depth 25 - Integrability of the central–elliptic twisted kernel against an idele character
AutomorphicForm.integrableOn_setIntegral_mul_finsum_centralElliptic_twistedAdelicKernel39 below · cited by 1 · depth 25 - Godement section: convergence, holomorphy and continuity for Re s>0
AutomorphicForm.integrable_and_differentiableAt_and_continuous_godementSection_of_mem_schwartzBruhat24 below · cited by 3 · depth 25 - Complex-place Casimirs of a factorizable test function: level and types
AutomorphicForm.isFactorizableTestFn_leftCasimirComplex_and_rightConv_mem_of_isArchBiFinite12 below · cited by 1 · depth 25 - Godement sections lie in the induced principal series
AutomorphicForm.isInducedSection_godementSection_of_forall_coe_eq_ideleNorm0 below · cited by 8 · depth 25 - Principal-level isotypic cusp forms lie in sum of cuspidal constituents
AutomorphicForm.isotypicCuspSubmodule_principal_inf_archCutSubmodule_le_iSup_isCuspConstituent329 below · cited by 2 · depth 25 - Non-degenerate invariant pairing forces dual K-types
AutomorphicForm.le_iSup_typeSubmodule_dual_of_invariant_pairing1 below · cited by 1 · depth 25 - Left-flow Casimir equals Casimir at a complex place
AutomorphicForm.leftCasimirComplex_eq_archCasimirAtComplex_of_isArchSmoothAtComplex1 below · cited by 4 · depth 25 - Finiteness of the σ-twisted scalar-class kernel integral
AutomorphicForm.lintegral_lintegral_tsum_enorm_twistedKernel_identityFamily_lt_top85 below · cited by 1 · depth 25 - Twisted norm of a scalar idele in odd-degree cyclic descent
AutomorphicForm.mem_range_idelicNorm_of_isNormOf_centralScalar_of_odd3 below · cited by 1 · depth 25 - Rankin–Selberg unfolding on GL₂ with mixed majorant
AutomorphicForm.peterssonIntegral_mul_bruhatEisenstein_eq_integral_whittakerCoefficient_mul_conj_rationalCentreUnipotentQuotient_of_integrable37 below · cited by 1 · depth 25 - Twisted elliptic transfer identity for one norm class
AutomorphicForm.setIntegral_mul_setIntegral_sigmaCentralizerDomain_eq_mul_sum_setIntegral_range_idelicNorm_of_normClassMap_eq_of_areMatchingOn269 below · cited by 1 · depth 25 - Central-norm twisted term in prime-degree cyclic base change
AutomorphicForm.setIntegral_sigmaCentralizerDomain_eq_mul_apply_centralScalar_of_normClassMap_eq_mk_scalar_of_central_transfer117 below · cited by 1 · depth 25 - Bruhat unfolding of a Godement section into an Epstein integral
AutomorphicForm.summable_godementSection_and_bruhatSeries_eq_mul_setIntegral_tsum_of_lintegral_tsum_enorm_lt_top4 below · cited by 4 · depth 25 - Absolute summability of cut traces over Siegel-pinned cusp classes
AutomorphicForm.summable_norm_cutTrace_of_isUnitFactorizableOfTypeAt_of_coversModCentre_of_subset96 below · cited by 1 · depth 25 - Absolute majorisation of the Bruhat series of a Godement section
AutomorphicForm.summable_norm_godementSection_bruhat_and_norm_add_tsum_norm_le_mul_setIntegral_tsum_norm_of_lintegral_tsum_enorm_lt_top4 below · cited by 2 · depth 25 - Leading term at s=1/2 unchanged by a local Weyl translation
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_localWeyl_sub_nhds_zero_of_flat_family69 below · cited by 1 · depth 25 - Intertwining residue unchanged by isometry at one archimedean place
AutomorphicForm.tendsto_sub_one_half_mul_weylIntertwiningIntegral_sub_nhds_zero_of_flat_family_of_archSupportedAt76 below · cited by 1 · depth 25 - Whittaker coefficients of a flat unitary Eisenstein family along the torus
AutomorphicForm.whittakerCoefficient_bruhatEisenstein_diagOne_eq_cpowChar_mul_sum_eulerProduct_of_flat_family_of_unitary58 below · cited by 3 · depth 25 - Whittaker coefficient of a conjugated, determinant-twisted automorphic function
AutomorphicForm.whittakerCoefficient_inv_star_mul_apply_det_eq_star_whittakerCoefficient_mul0 below · cited by 2 · depth 25 - Finite Haar volume of a determinant slab inside a twisted-centraliser tube
AutomorphicForm.adelicGLHaar_inter_setOf_inv_mul_sigmaAdelicAct_mem_center_mul_lt_top_of_forall_smul_inter76 below · cited by 1 · depth 26 - Four-cell decomposition of the adelic kernel
AutomorphicForm.adelicKernel_eq_four_parts_of_localFiniteness0 below · cited by 3 · depth 26 - Vanishing at non-norm central ideles for matching test functions
AutomorphicForm.apply_centralScalar_eq_zero_of_not_exists_isNormOf_of_isUnitFactorization_of_prime41 below · cited by 1 · depth 26 - Bruhat–Möbius relation at one archimedean place
AutomorphicForm.apply_weylInv_unipotent_mul_archSupportedAt_eq_norm_cpow_mul_apply2 below · cited by 1 · depth 26 - Bruhat relation at one finite place for induced sections
AutomorphicForm.apply_weylInv_unipotent_mul_localWeyl_eq_modulus_cpow_mul_apply3 below · cited by 1 · depth 26 - Galois action preserves archimedean height and finite integrality
AutomorphicForm.archHeight_glArch_sigmaAdelicAct_and_glFin_sigmaAdelicAct_mem_finiteIntegralGL20 below · cited by 5 · depth 26 - Continuity and compact support of truncated elliptic orbital integrals
AutomorphicForm.continuous_and_hasCompactSupport_setIntegral_fundamentalDomain_conj_centralScalar_mul_of_mem_ellipticCell20 below · cited by 1 · depth 26 - Strong multiplicity one for cuspidal constituents at principal level
AutomorphicForm.eq_of_isCuspConstituent_of_cuspConstituentMeets_principal_of_coversModCentre243 below · cited by 3 · depth 26 - Analytic continuation of the Eisenstein and intertwining families
AutomorphicForm.exists_analyticOnNhd_axis_continuation_bruhatEisenstein_weylIntertwiningIntegral_of_isArchKFinite_family207 below · cited by 9 · depth 26 - Truncated σ-twisted spectral identity along Hecke words
AutomorphicForm.exists_atomic_forall_tendsto_setIntegral_lambdaT_finsum_integral_sigmaAdelicAct_centralScalar_sub_twistedCutTrace_sub1,332 below · cited by 1 · depth 26 - Large-R limit: slope, summable atoms, small functional
AutomorphicForm.exists_atomic_forall_tendsto_tsum_integral_prod_pow_mul_affine_oscillatory_sub_mul_of_placewise_bound_of_sum_lipschitz1 below · cited by 2 · depth 26 - Twisted centralizer of a σ-conjugate scalar is GL₂(A)
AutomorphicForm.exists_continuousMulEquiv_centralizer_twistedCentralizer_of_eq_scalar0 below · cited by 6 · depth 26 - Existence of a unit factorisation at S with prescribed factors
AutomorphicForm.exists_continuous_hasCompactSupport_isUnitFactorization_and_union_of_isArchTestFactor_of_isLocalTestFn1 below · cited by 3 · depth 26 - Truncated hyperbolic terms compared with a uniform slope λ
AutomorphicForm.exists_continuous_noAtomicMass_integrableOn_and_hyperbolicTerm_sub_finrank_mul_const_mul_sum_eq_of_areMatchingAt_uniform1,509 below · cited by 1 · depth 26 - Matched unipotent terms: affine in R with atom-free remainder
AutomorphicForm.exists_continuous_noAtomicMass_integrableOn_and_unipotentTerm_sub_const_mul_sum_eq_of_areMatchingAt362 below · cited by 1 · depth 26 - Countable complete orthonormal flat families of induced sections
AutomorphicForm.exists_countable_orthonormal_flat_isInducedSection_family_complete_principalLevel_archCutSubmodule18 below · cited by 2 · depth 26 - Twisted principal-series Hecke table is an Eisenstein table
AutomorphicForm.exists_eisensteinTableOf_eq_table_of_isUnitaryChar_of_isUnramifiedCharAt7 below · cited by 2 · depth 26 - Maass–Selberg relation on the unitary axis, flat families
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_axis_continuation_mul_conj_eq_maassSelberg_or_twoTerm_slab_of_flat271 below · cited by 4 · depth 26 - Truncated parabolic term splits into hyperbolic and unipotent cells
AutomorphicForm.exists_forall_le_integrableOn_and_setIntegral_parabolic_eq_hyperbolicCell_add_unipotentCell94 below · cited by 2 · depth 26 - Hyperbolic–unipotent splitting of the truncated twisted parabolic term
AutomorphicForm.exists_forall_le_integrableOn_and_setIntegral_twistedParabolic_eq_hyperbolicCell_add_unipotentCell166 below · cited by 1 · depth 26 - Integrability of the ξ-folded truncated twisted GL₂ kernel
AutomorphicForm.exists_forall_le_integrableOn_mul_lambdaT_twistedAdelicKernel_canonicalTruncationDomain_prod86 below · cited by 1 · depth 26 - Integrability of the centre-folded truncated GL₂ adelic kernel
AutomorphicForm.exists_forall_le_integrableOn_setIntegral_mul_lambdaT_adelicKernel_of_isTruncationDatum82 below · cited by 1 · depth 26 - Bounding the affine coefficients of the hyperbolic twisted term
AutomorphicForm.exists_forall_norm_add_norm_le_of_forall_setIntegral_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_affine413 below · cited by 1 · depth 26 - Modulus of an idele class character is a power of the norm
AutomorphicForm.exists_forall_norm_apply_eq_ideleNorm_rpow_of_continuous_of_trivial5 below · cited by 4 · depth 26 - Truncated twisted unipotent term as weighted Hecke-word moments
AutomorphicForm.exists_forall_setIntegral_finsum_unipotentCell_sub_indicator_constantTerm_eq_weighted_moments_unram254 below · cited by 2 · depth 26 - Integrated continuous-spectrum identity for the truncated GL₂ kernel
AutomorphicForm.exists_forall_setIntegral_lambdaT_finsum_sub_lambdaT_tsum_sub_lambdaT_finsum_chiDet_eq_mul_integral_sum_rightConv_mul_setIntegral_lambdaT_axis_continuation1,261 below · cited by 1 · depth 26 - Transfer data at a twisted class with elliptic norm
AutomorphicForm.exists_haar_sigmaCentralizer_centralizer_covolume_and_twistedOrbital_eq_of_normClassMap_eq_of_areMatchingOn32 below · cited by 1 · depth 26 - Idelic base change for a cyclic extension: fixed idèles form the image
AutomorphicForm.exists_idelesBaseChange_continuous_injective_norm_pow_range_eq_fixed4 below · cited by 2 · depth 26 - Central twisted norm forces twisted conjugacy to a scalar (odd degree)
AutomorphicForm.exists_inv_mul_mul_map_eq_scalar_of_sigmaNormPow_eq_map_scalar_of_odd1 below · cited by 1 · depth 26 - Properness modulo the centre of the twisted orbit map on GL₂(A_L)
AutomorphicForm.exists_isCompact_forall_exists_inv_mul_sigmaAdelicAct_mem_center_of_mem_center_mul16 below · cited by 2 · depth 26 - Bounded-height part of a centre-cut Siegel set lies in a compact
AutomorphicForm.exists_isCompact_forall_mem_centreCutSiegelSet_archHeight_le_mem1 below · cited by 4 · depth 26 - Haar normalisation on the centralizer of an adelic scalar in GL₂
AutomorphicForm.exists_isHaarMeasure_centralizer_forall_isFundamentalDomain_op_inter_eq_mul_log_and_isOrbitalIntegralOn_centralScalar_iff22 below · cited by 2 · depth 26 - Twisted section functions exist at scalar twisted classes
AutomorphicForm.exists_isTwistedSectionFnOn_adeleRing_of_isSigmaConjugate_scalar63 below · cited by 1 · depth 26 - Logarithmic band covolume in a twisted centralizer of a scalar class
AutomorphicForm.exists_measure_fundamentalDomain_op_twistedCentralizer_inter_ideleNorm_det_Icc_eq_mul_log_of_eq_scalar27 below · cited by 1 · depth 26 - Idelic norms are norm strings of adelic scalar matrices
AutomorphicForm.exists_normString_scalar_eq_toTensorGL_centralScalar_of_mem_range_idelicNorm3 below · cited by 1 · depth 26 - Shaped raw cusp vector over ℚ with unit-shell support
AutomorphicForm.exists_shapedRawVector_finWhittaker_support_transl_rat135 below · cited by 1 · depth 26 - Cuspidal realization transfers to the standard Siegel window
AutomorphicForm.exists_smoothCuspRealizationAt_productionPinsGeneral_toFun_eq_of_lt_of_coversModCentre24 below · cited by 1 · depth 26 - Summable dominant for continuous-spectrum Maass–Selberg pairings
AutomorphicForm.exists_summable_dominant_rightConv_axis_family_maassSelberg_pairings_of_isUnitFactorization_sum_lipschitz414 below · cited by 1 · depth 26 - Asymptotically affine truncated parabolic term, unit-factorizable f
AutomorphicForm.exists_tendsto_setIntegral_lambdaT_adelicKernel_sub_centralElliptic_sub_affine_atTop_of_isUnitFactorization399 below · cited by 1 · depth 26 - Finiteness of central and elliptic terms for GL₂
AutomorphicForm.finite_sep_exists_apply_inv_mul_globalPoints_mul_centralScalar_mul_ne_zero_of_hasCompactSupport12 below · cited by 1 · depth 26 - Finiteness of continuous idele class characters of square ξ and level N
AutomorphicForm.finite_setOf_squaresToXi_continuous_apply_det_eq_one_of_mem_principalLevel5 below · cited by 3 · depth 26 - Cuspidal class contribution equals its cut trace, principal level
AutomorphicForm.finsum_setIntegral_convOp_mul_conj_eq_cutTrace_of_orthonormal_principalLevel_of_isFundamentalDomain_slab22 below · cited by 1 · depth 26 - Affine dependence of the hyperbolic term on the truncation parameter
AutomorphicForm.forall_exists_setIntegral_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_affine_bare212 below · cited by 3 · depth 26 - Cuspidal kernel: locally uniform bounds and class-wise integrability
AutomorphicForm.forall_isCompact_exists_tsum_norm_convOp_mul_conj_le_and_summable_setIntegral_norm_finsum_of_orthonormal_principalLevel_of_isFundamentalDomain_slab503 below · cited by 3 · depth 26 - Integrability of the central–elliptic twisted kernel over centre-cut Siegel translates
AutomorphicForm.integrableOn_iUnion_centreCutSiegelSet_setIntegral_mul_finsum_borel_centralElliptic22 below · cited by 1 · depth 26 - Hecke words extracted from int f·(χ∘det)
AutomorphicForm.integral_mul_chiDet_eq_prod_pow_mul_pow_mul_integral_mul_chiDet_of_isUnitFactorization0 below · cited by 1 · depth 26 - Self-adjointness of M(0) on flat sections, case μ=ν
AutomorphicForm.integral_mul_conj_axis_continuation_weylIntertwiningIntegral_zero_eq_of_eq_of_flat272 below · cited by 2 · depth 26 - Twisting an induced section by ‖det‖^{w/2}
AutomorphicForm.isInducedSection_mul_cpowChar_and_continuous_and_maximalCompactAway_of_isInducedSection_of_principalLevel4 below · cited by 2 · depth 26 - Twisting by a complex power of the idelic modulus preserves unramifiedness
AutomorphicForm.isUnramifiedCharAt_mul_cpowChar_of_isUnramifiedCharAt2 below · cited by 2 · depth 26 - Induced sections of level N force characters unramified outside N
AutomorphicForm.isUnramifiedCharAt_of_isInducedSection_etaFst_etaSnd_of_ne_zero_of_principalLevel3 below · cited by 3 · depth 26 - Finite-dimensional Hecke-stable cusp spaces lie in isotypic sums
AutomorphicForm.le_iSup_isotypicCuspSubmodule_principal_inf_archCutSubmodule_of_finiteDimensional_of_forall_heckeCosetSum_mem17 below · cited by 1 · depth 26 - GL₂(L⊗_KA_K)≅ GL₂(A_L) respects Galois action and embeddings
AutomorphicForm.map_genuineRingEquiv_sigmaGL_and_toTensorGL_and_includeLeft1 below · cited by 15 · depth 26 - Adelic matching at σ-classes with central norm, prime degree
AutomorphicForm.mul_eq_mul_of_isTwistedOrbitalIntegralOn_of_isOrbitalIntegralOn_centralScalar_of_areMatchingLocal696 below · cited by 1 · depth 26 - Topological and Haar side conditions for GL₂(Kᵥ) and N₂
AutomorphicForm.secondCountableTopology_and_locallyCompactSpace_gl_two_and_isClosed_range_unipotentGL2Hom0 below · cited by 28 · depth 26 - Scalar stabilising γ₀ leaves the centralizer-domain integral unchanged
AutomorphicForm.setIntegral_fundamentalDomain_conj_centralScalar_mul_eq_of_scalar_mul_eq_conj9 below · cited by 1 · depth 26 - Vanishing of a truncated elliptic orbital term for GL₂
AutomorphicForm.setIntegral_fundamentalDomain_conj_centralScalar_mul_eq_zero_of_forall_isOrbitalIntegralOn_eq_zero20 below · cited by 1 · depth 26 - Slab integral of f(x⁻¹γ x): covolume times orbital integral
AutomorphicForm.setIntegral_fundamentalDomain_slab_eq_measureReal_mul_of_isOrbitalIntegralOn12 below · cited by 3 · depth 26 - Twisted slab identity: covolume times twisted orbital integral
AutomorphicForm.setIntegral_fundamentalDomain_slab_sigmaAdelicAct_eq_measureReal_mul_integral_map_of_isTwistedSectionFnOn18 below · cited by 2 · depth 26 - Twisted slab identity for one twisted class in GL₂
AutomorphicForm.setIntegral_fundamentalDomain_slab_sigmaCentralizer_eq_measureReal_mul_integral_of_forall_exists_mem_center7 below · cited by 2 · depth 26 - Quadratic base change: unfolded twisted term at a non-scalar class
AutomorphicForm.setIntegral_sigmaCentralizerDomain_eq_mul_apply_centralScalar_of_normClassMap_eq_mk_scalar_of_forall_ne_scalar_of_finrank_eq_two62 below · cited by 1 · depth 26 - Unitary principal-series tables lie in the ξ-box
AutomorphicForm.table_axis_mem_setOf_xiBox_of_isUnitaryChar_of_mul_mul_rpow_eq0 below · cited by 2 · depth 26 - Unitary twist transports the shaped Whittaker package over ℚ
AutomorphicForm.unitaryTwist_transport_shapedRawVector_transl_rat102 below · cited by 1 · depth 26 - σ-invariant idele characters agree at Hecke generators above v
AutomorphicForm.apply_det_heckeGen_eq_of_asIdeal_eq_smul_of_sigmaInvariant_unram6 below · cited by 1 · depth 27 - Central transfer at scalars above a finite place
AutomorphicForm.areMatchingLocal_central_transfer_and_eq_zero_of_not_exists_isNormOf31 below · cited by 3 · depth 27 - Left GL₂(F)-invariance of the continued Eisenstein series
AutomorphicForm.axis_continuation_bruhatEisenstein_globalPoints_mul_eq_of_isArchKFinite_family6 below · cited by 12 · depth 27 - Regularity and Cauchy–Schwarz bounds for flat Maass–Selberg pairings
AutomorphicForm.continuous_and_hasDerivAt_axis_continuation_weylIntertwiningIntegral_pairings_of_flat0 below · cited by 2 · depth 27 - Continuity in t of K-coefficients of πᵢₜ(f)
AutomorphicForm.continuous_integral_rightConv_axis_mul_conj_of_isArchKFinite_family2 below · cited by 4 · depth 27 - Right convolution preserves the cut isotypic cuspidal space
AutomorphicForm.convOp_mem_isotypicCuspSubmodule_inf_archCutSubmodule_principalLevel_of_isBiInvariantUnder_of_isFundamentalDomain_slab21 below · cited by 8 · depth 27 - Principal- and level-one pins give the same K-finite cuspidal space
AutomorphicForm.cuspKFiniteSubmodule_productionPinsOf_principalLevel_eq_levelOne0 below · cited by 1 · depth 27 - Matching transports central translates along the idelic norm
AutomorphicForm.eq_comp_idelicNorm_of_isTwistedOrbitalIntegralOn_centralScalar_mul_of_isOrbitalIntegralOn_centralScalar_mul_of_areMatchingOn4 below · cited by 2 · depth 27 - Continuation of the non-constant part of the GL₂ Eisenstein family
AutomorphicForm.exists_analyticOnNhd_continuation_bruhatEisenstein_sub_constantTerm_of_re_nonneg_of_isArchKFinite_family153 below · cited by 1 · depth 27 - Continuation of the GL₂ intertwining integral off one point
AutomorphicForm.exists_analyticOnNhd_continuation_weylIntertwiningIntegral_of_re_nonneg_of_isArchKFinite_family138 below · cited by 2 · depth 27 - Spectral side of the σ-twisted trace formula along Hecke words
AutomorphicForm.exists_atomic_forall_tendsto_setIntegral_lambdaT_finsum_integral_sigmaAdelicAct_centralScalar_sub_tsum_finsum_setIntegral_twistedConvOp_sub1,331 below · cited by 1 · depth 27 - Uniform bounds and summability for adelic GL₂ Eisenstein data
AutomorphicForm.exists_bound_card_and_archParam_weight_and_summable_of_orthonormal_flat_isInducedSection_family40 below · cited by 1 · depth 27 - Boundedness of the unitarised finite Whittaker factor over ℚ
AutomorphicForm.exists_bound_finWhittaker_mul_ideleNorm_det_rpow_of_isCuspAutomorphicFnAt_rat77 below · cited by 1 · depth 27 - Pushing a torus functional to the table space along Hecke words
AutomorphicForm.exists_clm_cylinder_noAtomicMass_and_apply_monomial_eq_sum_laurentCoeff_mul_of_box_noAtomicMass4 below · cited by 2 · depth 27 - Affine shape of base-changed unipotent terms along Hecke words
AutomorphicForm.exists_clm_noAtomicMass_forall_sum_slotFamilyCoeff_mul_setIntegral_unipotentCell_eq_mul_add277 below · cited by 1 · depth 27 - A uniform transfer constant in the hyperbolic-term base-change comparison
AutomorphicForm.exists_const_forall_exists_windingDatum_integrableOn_and_hyperbolicTerm_sub_finrank_mul_const_mul_sum_eq_mul_sum_coeff_add_sum_coeff_of_areMatchingAt1,495 below · cited by 1 · depth 27 - Twisted centralizer of δ versus centralizer of its norm
AutomorphicForm.exists_continuousMulEquiv_twistedCentralizer_centralizer_coupled_of_isNormRep0 below · cited by 1 · depth 27 - Elliptic orbital integrals of central translates, continuous and compactly supported in u
AutomorphicForm.exists_continuous_hasCompactSupport_forall_isOrbitalIntegralOn_mul_centralScalar_of_mem_ellipticCell5 below · cited by 3 · depth 27 - Countably many continuous unitary idele class characters on A_K¹
AutomorphicForm.exists_countable_family_isUnitaryChar_isIdeleClassChar_forall_exists_eqOn_normOneIdeles5 below · cited by 1 · depth 27 - Hilbert's Theorem 90 for GL₂ over adeles
AutomorphicForm.exists_eq_inv_mul_sigmaAdelicAct_of_prod_sigmaAdelicAct_pow_eq_one4 below · cited by 1 · depth 27 - Dominated and integrable continuous spectral kernel after truncation
AutomorphicForm.exists_forall_dominated_sum_rightConv_axis_continuation_and_integrable_prod_lambdaT443 below · cited by 2 · depth 27 - Pointwise spectral identity for the GL₂ kernel on the unitary axis
AutomorphicForm.exists_forall_finsum_integral_centralScalar_sub_tsum_convOp_sub_finsum_chiDet_eq_mul_integral_sum_rightConv_axis_continuation1,238 below · cited by 2 · depth 27 - Integrability of truncated axis-continued Eisenstein products on Φ₀
AutomorphicForm.exists_forall_integrableOn_axis_continuation_mul_conj_lambdaT_canonicalTruncationDomain143 below · cited by 1 · depth 27 - Integrability of the truncated hyperbolic and unipotent kernels
AutomorphicForm.exists_forall_le_integrableOn_hyperbolicCell_and_unipotentCell_sub_indicator_constantTerm91 below · cited by 2 · depth 27 - Integrability of the truncated twisted hyperbolic and unipotent kernels
AutomorphicForm.exists_forall_le_integrableOn_twistedHyperbolicCell_and_twistedUnipotentCell_sub_indicator_constantTerm162 below · cited by 2 · depth 27 - Pointwise cell decomposition of the truncated GL₂ adelic kernel
AutomorphicForm.exists_forall_le_lambdaT_adelicKernel_eq_centralElliptic_add_unipotentCell_add_hyperbolicCell4 below · cited by 1 · depth 27 - Uniform polynomial bound for the GL₂ scattering derivative on the unitary axis
AutomorphicForm.exists_forall_lintegral_norm_deriv_axis_continuation_weylIntertwiningIntegral_le_mul_pow_archParam_weight391 below · cited by 3 · depth 27 - Uniform bound for the truncated twisted GL₂ kernel on Siegel translates
AutomorphicForm.exists_forall_norm_lambdaT_twistedAdelicKernel_centralScalar_mul_le_of_subset_centreCutSiegelSet_translates70 below · cited by 2 · depth 27 - Polynomial sup-norm bound on compacta for Casimir-eigen cusp forms
AutomorphicForm.exists_forall_norm_le_mul_rpow_mul_eLpNorm_of_mem_isotypicCuspSubmodule_principal_of_archCasimir_eq_smul_of_isCompact391 below · cited by 2 · depth 27 - Uniform rapid decay of K-matrix coefficients on the unitary axis
AutomorphicForm.exists_forall_norm_rightConv_axis_pairing_add_norm_deriv_le_mul_rpow_neg_archParam_of_isUnitFactorization20 below · cited by 2 · depth 27 - Hyperbolic term of the twisted trace formula: affine or zero
AutomorphicForm.exists_forall_setIntegral_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_mul_sum_orbital_add_sum_weightedOrbital_or_eq_zero_of_isFactorizableTestFn211 below · cited by 3 · depth 27 - Truncated twisted unipotent term along Hecke words via local zetas
AutomorphicForm.exists_forall_setIntegral_finsum_unipotentCell_sub_indicator_constantTerm_eq_mul_localZeta_twistedLocalFactor_unram245 below · cited by 1 · depth 27 - Maass–Selberg relations on the unitary axis at distinct parameters
AutomorphicForm.exists_forall_setIntegral_lambdaT_axis_continuation_mul_conj_eq_maassSelberg_and_eq_twoTerm_slab_of_ne267 below · cited by 3 · depth 27 - Hecke-word bound for summed twisted and weighted orbital integrals
AutomorphicForm.exists_forall_sum_integral_norm_orbital_add_weightedOrbital_le_of_isSemiLocalFactorization287 below · cited by 1 · depth 27 - Casimir-weighted Hilbert–Schmidt bound for level-N convolution on cusp forms
AutomorphicForm.exists_forall_sum_rpow_mul_sqrt_sum_eLpNorm_convOp_sq_le_of_orthonormal_isotypicCuspSubmodule_principal_of_archCasimir_eq_smul371 below · cited by 2 · depth 27 - Translates of a centre-cut Siegel set lie in a determinant slab
AutomorphicForm.exists_iUnion_image_mul_centreCutSiegelSet_subset_setOf_ideleNorm_det_mem_Icc5 below · cited by 2 · depth 27 - Compactly controlled Hilbert 90 for GL₂ over the adeles
AutomorphicForm.exists_isCompact_setOf_inv_mul_sigmaGL_mem_subset_twistedCentralizer_one_mul59 below · cited by 1 · depth 27 - Properness of the twisted orbit map modulo the twisted centralizer
AutomorphicForm.exists_isCompact_setOf_twistedConj_mem_subset_twistedCentralizer_mul_of_forall_ne_scalar_of_finrank_eq_two6 below · cited by 1 · depth 27 - Shell vanishing forces compact support and unit idele norm
AutomorphicForm.exists_isCompact_support_and_ideleNorm_det_eq_one_of_shellSupport_rat5 below · cited by 1 · depth 27 - A common coset system for U₁(N) and K(N) at v ∤ N
AutomorphicForm.exists_isHeckeCosetSystem_levelOne_and_principalLevel_heckeGen_of_not_dvd1 below · cited by 1 · depth 27 - Twisted section functions exist at classes with regular semisimple norm
AutomorphicForm.exists_isTwistedSectionFnOn_adeleRing_of_isRegularSemisimple_normString9 below · cited by 4 · depth 27 - Uniform finite volume bound on determinant slabs in the twisted locus
AutomorphicForm.exists_lt_top_forall_measure_preimage_le_of_isHaarMeasure_eqLocus_sigmaAdelicAct_center75 below · cited by 1 · depth 27 - Log-linearity of norm-band covolumes for elliptic centralizer tori
AutomorphicForm.exists_measure_fundamentalDomain_centralizer_inter_ideleNorm_det_Icc_eq_mul_log_of_mem_ellipticCell8 below · cited by 1 · depth 27 - Log-linear band volume for GL₂(K) in a central centralizer
AutomorphicForm.exists_measure_fundamentalDomain_op_centralizer_inter_ideleNorm_det_Icc_eq_mul_log_of_mem_center21 below · cited by 1 · depth 27 - Log-linear band covolume for twisted centralizers in degree two
AutomorphicForm.exists_measure_fundamentalDomain_op_twistedCentralizer_inter_ideleNorm_det_Icc_eq_mul_log_of_forall_ne_scalar_of_finrank_eq_two29 below · cited by 2 · depth 27 - Adelic norm string of a scalar from local data, odd prime degree
AutomorphicForm.exists_normString_scalar_eq_toTensorGL_centralScalar_of_forall_of_finrank_ne_two8 below · cited by 1 · depth 27 - Orthonormal complete cusp system at level N on a slab
AutomorphicForm.exists_orthonormal_isotypicCuspSubmodule_levelOne_of_isFundamentalDomain_slab340 below · cited by 3 · depth 27 - Orthonormal basis for the maximal compact K-pairing
AutomorphicForm.exists_orthonormal_maximalCompactHaar_basis_of_finiteDimensional0 below · cited by 1 · depth 27 - Self-adjointness of truncation on the canonical truncation domain
AutomorphicForm.exists_pos_forall_setIntegral_lambdaT_mul_conj_eq_setIntegral_lambdaT_mul_conj_lambdaT_canonicalTruncationDomain39 below · cited by 4 · depth 27 - Mean-square strong multiplicity one on an ample Siegel window
AutomorphicForm.exists_setLIntegral_sub_sum_translate_sq_lt_of_agreesAwayFromFinite_of_coversModCentre_ample_principal136 below · cited by 1 · depth 27 - Simultaneous unit-shell shaping at all primes of S
AutomorphicForm.exists_shapedRaw_bundle_forall_shellSupport_transl_rat115 below · cited by 1 · depth 27 - Affine asymptotics of the truncated hyperbolic term, unit factorisation
AutomorphicForm.exists_tendsto_setIntegral_hyperbolicCell_sub_affine_atTop_of_isUnitFactorization224 below · cited by 1 · depth 27 - Affine asymptotics of the truncated unipotent term, unit-factorizable f
AutomorphicForm.exists_tendsto_setIntegral_unipotentCell_sub_affine_atTop_of_isUnitFactorization263 below · cited by 1 · depth 27 - Finite-dimensionality of K-finite induced sections at principal level
AutomorphicForm.finiteDimensional_span_setOf_isInducedSection_principalLevel_archCutSubmodule4 below · cited by 1 · depth 27 - Cuspidal class contribution equals its twisted cut trace
AutomorphicForm.finsum_setIntegral_twistedConvOp_mul_conj_eq_twistedCutTrace_of_orthonormal_of_isFundamentalDomain_slab38 below · cited by 2 · depth 27 - Archimedean Casimir operators act by scalars on cut isotypic cusp spaces
AutomorphicForm.forall_mem_cuspClasses_exists_forall_isArchSmoothAt_and_archCasimirAt_eq_smul_of_mem_isotypicCuspSubmodule_principal_inf_archCutSubmodule365 below · cited by 2 · depth 27 - Integrability and class-wise summability of the cuspidal diagonal kernel
AutomorphicForm.integrableOn_convOp_mul_conj_and_summable_setIntegral_norm_finsum_convOp_mul_conj_of_orthonormal_principalLevel_of_isFundamentalDomain_slab371 below · cited by 2 · depth 27 - Integrability and positive mass of |W_f|² on the cut
AutomorphicForm.integrable_indicator_normSq_and_measure_ne_zero_of_isCompact_support_rat16 below · cited by 1 · depth 27 - Integrability of the cuspidal kernel along unipotent orbits
AutomorphicForm.integrable_tsum_convOp_mul_conj_unipotentGL2_mul_of_orthonormal_principalLevel_of_isFundamentalDomain_slab505 below · cited by 1 · depth 27 - Unitarity of the normalised Weyl intertwining operator on the unitary axis
AutomorphicForm.integral_axis_continuation_weylIntertwiningIntegral_mul_conj_eq_integral_mul_conj_of_isUnitaryChar270 below · cited by 13 · depth 27 - Compactness of the principal level meeting the finite-adelic subgroup
AutomorphicForm.isCompact_principalLevel_inf_finiteAdelicGL2Subgroup1 below · cited by 5 · depth 27 - Cuspidal constituents agree for principal- and level-one pins
AutomorphicForm.isCuspConstituent_productionPinsOf_principalLevel_iff_levelOne0 below · cited by 2 · depth 27 - Twisting a GL₂ test function by ‖det‖^{w/2}
AutomorphicForm.isFactorizableTestFn_and_isBiInvariantUnder_and_isArchBiFinite_mul_ideleNorm_det_rpow5 below · cited by 3 · depth 27 - Unimodularity of the σ-twisted centraliser modulo the centre
AutomorphicForm.isMulRightInvariant_of_isHaarMeasure_eqLocus_sigmaAdelicAct_center6 below · cited by 2 · depth 27 - Unimodularity of the adelic twisted centraliser with central norm
AutomorphicForm.isMulRightInvariant_twistedCentralizer_adeleRing_of_normString_eq_toTensorGL_centralScalar_of_finrank_eq_two13 below · cited by 4 · depth 27 - Openness of the principal level K(N) in GL₂(A_F)
AutomorphicForm.isOpen_principalLevel0 below · cited by 13 · depth 27 - Unitary twist by ‖det‖^{-σ₀/2} preserves rapid decay on Siegel sets
AutomorphicForm.isRapidlyDecreasingOnSiegelSets_mul_ideleNorm_det_rpow_of_isCuspAutomorphicFnAt_rat83 below · cited by 1 · depth 27 - Isotypic cusp forms of zero principal level vanish
AutomorphicForm.isotypicCuspSubmodule_principal_bot_eq_bot_of_productionPinsOf0 below · cited by 1 · depth 27 - Isotypic cuspidal spaces vanish on a non-positive Siegel window
AutomorphicForm.isotypicCuspSubmodule_principal_eq_bot_of_nonpos1 below · cited by 1 · depth 27 - Raising the determinant floor does not enlarge isotypic cusp spaces
AutomorphicForm.isotypicCuspSubmodule_principal_le_isotypicCuspSubmodule_principal_of_le_of_ne_bot4 below · cited by 1 · depth 27 - Finiteness of |det|^t over norm balls in GL₂(ℚₚ)
AutomorphicForm.lintegral_indicator_norm_le_mul_norm_det_rpow_lt_top22 below · cited by 2 · depth 27 - Haar measure on local GL₂ is invariant under g↦ ^tg⁻¹
AutomorphicForm.map_transposeInvN_eq_self_of_isHaarMeasure_fin_two0 below · cited by 4 · depth 27 - L² continuity in s of truncated Eisenstein families
AutomorphicForm.memLp_two_lambdaT_and_tendsto_eLpNorm_lambdaT_sub_restrict_canonicalTruncationDomain_of_axis_continuation_family133 below · cited by 8 · depth 27 - Conjugation-invariance of the four GL₂ cells
AutomorphicForm.mem_cells_iff_of_isConj0 below · cited by 1 · depth 27 - K-finiteness of smoothed isotypic cusp forms at principal level
AutomorphicForm.mem_cuspKFiniteSubmodule_of_mem_isotypicCuspSubmodule_principal_inf_archCutSubmodule_of_rightConv_eq_smul85 below · cited by 1 · depth 27 - Global central transfer with coupled measures: c_K I' = c_L I
AutomorphicForm.mul_eq_mul_of_isTwistedOrbitalIntegralOn_of_isOrbitalIntegralOn_centralScalar_of_coupled116 below · cited by 1 · depth 27 - Global matching at central-norm classes of the second kind
AutomorphicForm.mul_eq_mul_of_isTwistedOrbitalIntegralOn_of_isOrbitalIntegralOn_centralScalar_of_forall_ne_scalar_of_finrank_eq_two689 below · cited by 1 · depth 27 - Twist transport and truncated diagonal of the residual kernel
AutomorphicForm.resKernel_twist_and_lambdaT_resKernel_diag21 below · cited by 1 · depth 27 - Non-σ-invariant idele character kills the truncated unipotent term
AutomorphicForm.setIntegral_finsum_unipotentCell_sub_indicator_constantTerm_eq_zero_of_not_sigmaInvariant_unram12 below · cited by 2 · depth 27 - Torus quotient in a determinant slab for GL₂(A_F)
AutomorphicForm.setIntegral_fundamentalDomain_slab_eq_measureReal_smul_integral_of_forall_integral_eq_one6 below · cited by 2 · depth 27 - Twist-invariance of summed cut cuspidal traces on GL₂
AutomorphicForm.tsum_cutTrace_eq_tsum_cutTrace_mul_ideleNorm_det_rpow_of_subset_slab14 below · cited by 1 · depth 27 - Integrable Whittaker slices for reproduced cusp forms over ℚ
AutomorphicForm.whittakerCoefficientIntegrable_of_isCuspAutomorphicFnAt_of_rightConv_eq_rat90 below · cited by 3 · depth 27 - Constant term commutes with continuation of an Eisenstein family
AutomorphicForm.analyticOnNhd_constantTerm_and_eq_add_of_axis_continuation_family4 below · cited by 9 · depth 28 - Base change of idele characters at Hecke generators
AutomorphicForm.apply_det_heckeGen_pow_inertiaDeg_eq_apply_det_heckeGen_of_comp_idelicNorm_of_unramified4 below · cited by 3 · depth 28 - Vanishing of a local test function at a scalar from nearby orbital integrals
AutomorphicForm.apply_scalar_eq_zero_of_nhds_forall_isRegularSemisimple_isOrbitalIntegral_eq_zero6 below · cited by 1 · depth 28 - Right convolution preserves Casimir eigenvalues at a complex place
AutomorphicForm.archCasimirAtComplex_rightConv_eq_smul_of_archCasimirAtComplex_eq_smul_of_isArchSmoothAtComplex_of_isFactorizableTestFn9 below · cited by 1 · depth 28 - Archimedean central transfer for matching archimedean test factors
AutomorphicForm.areMatchingArch_central_transfer_of_scalar77 below · cited by 1 · depth 28 - Smoothness and slab bounds for archimedean derivatives of cusp forms
AutomorphicForm.continuous_archDerivAt_and_exists_bound_slab_of_isIsotypicCuspFormAt_of_rightConv_eq_rat84 below · cited by 1 · depth 28 - Continuity and GL₂(K)-automorphy of the residual kernel
AutomorphicForm.continuous_uncurry_finsum_chiDet_mul_chiDet_inv_and_apply_globalPoints_mul_and_apply_centralScalar_mul0 below · cited by 3 · depth 28 - Continuity and equivariance of the centre-folded GL₂ kernel
AutomorphicForm.continuous_uncurry_finsum_integral_centralScalar_mul_apply_inv_mul_globalPoints_mul_centralScalar_mul5 below · cited by 3 · depth 28 - Joint continuity and automorphy of the cuspidal kernel
AutomorphicForm.continuous_uncurry_tsum_convOp_mul_conj_of_orthonormal_isotypicCuspSubmodule504 below · cited by 3 · depth 28 - Hecke coset sums commute with right convolution, principal level
AutomorphicForm.cosetSum_rightConv_of_isLevelSphericalOfType_principal1 below · cited by 1 · depth 28 - Countable index set for orthonormal adelic cusp forms
AutomorphicForm.countable_index_of_orthonormal_isotypicCuspSubmodule_principalLevel_of_isFundamentalDomain_slab2 below · cited by 1 · depth 28 - Residual χ∘det block of the σ-twisted spectral side
AutomorphicForm.exists_atomic_forall_integrableOn_and_tendsto_setIntegral_lambdaT_finsum_twistedConvOp_chiDet_mul_chiDet_inv29 below · cited by 1 · depth 28 - Continuous block of the σ-twisted spectral side along Hecke words
AutomorphicForm.exists_atomic_forall_tendsto_setIntegral_lambdaT_finsum_integral_sigmaAdelicAct_sub_lambdaT_tsum_finsum_twistedConvOp_sub_lambdaT_finsum_twistedConvOp_chiDet_sub1,325 below · cited by 1 · depth 28 - A uniform transfer constant in the twisted hyperbolic comparison
AutomorphicForm.exists_const_forall_exists_windingDatum_sub_finrank_mul_const_mul_sum_eq_sum_mul_coeff_of_hyperbolicTerm_eq_affine1,489 below · cited by 1 · depth 28 - Godement Eisenstein series: continuation, functional equation, strip bounds
AutomorphicForm.exists_entire_eq_godementEisenstein_fe_norm_le_of_mem_schwartzBruhat286 below · cited by 1 · depth 28 - Entire L-normalised Whittaker terms of the Bruhat–Eisenstein family
AutomorphicForm.exists_entire_whittakerCoefficient_bruhatEisenstein_continuation_summable_norm_tsum_le_rpow_neg_of_isArchKFinite_family_of_unitary93 below · cited by 1 · depth 28 - Finitely many local character possibilities at fixed principal level
AutomorphicForm.exists_finite_forall_isUnramifiedCharAt_and_localChar_eq_of_isInducedSection_etaFst_etaSnd_of_ne_zero_of_principalLevel6 below · cited by 8 · depth 28 - Near-integrality off a finite set of places on GL₂(L⊗_KA_K)
AutomorphicForm.exists_finset_mem_nhds_forall_tensorPlace_mem_semiLocalIntegralSet0 below · cited by 4 · depth 28 - Uniform weight bound for non-zero induced sections of listed type
AutomorphicForm.exists_forall_abs_weight_le_of_isInducedSection_ne_zero_archCutSubmodule12 below · cited by 4 · depth 28 - Almost-everywhere spectral expansion of the continuous kernel for GL₂
AutomorphicForm.exists_forall_ae_prod_restrict_canonicalTruncationDomain_finsum_integral_centralScalar_sub_tsum_convOp_sub_finsum_chiDet_eq_mul_tsum_integral_sum_rightConv_axis_continuation1,237 below · cited by 1 · depth 28 - Flat sections: intertwining integral as completed L-ratio with axis bounds
AutomorphicForm.exists_forall_completedL_mul_axis_continuation_weylIntertwiningIntegral_eq_mul_normalizedIntertwining_and_lintegral_le_of_flat350 below · cited by 1 · depth 28 - Summable integrable dominants for the GL₂ continuous spectral sum
AutomorphicForm.exists_forall_dominated_sum_rightConv_axis_continuation_of_isCompact439 below · cited by 1 · depth 28 - L² bounds for derivative words of Casimir eigenfunctions
AutomorphicForm.exists_forall_eLpNorm_foldr_archDeriv_le_mul_rpow_mul_eLpNorm_of_mem_archCutSubmodule_of_archCasimir_eq_smul62 below · cited by 1 · depth 28 - Compact-set L² bound by the truncation domain L² norm
AutomorphicForm.exists_forall_eLpNorm_restrict_le_mul_eLpNorm_restrict_canonicalTruncationDomain_of_isLsXiFunction20 below · cited by 9 · depth 28 - Maass–Selberg relation on the unitary axis, diagonal case μ=ν
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_axis_continuation_mul_conj_eq_maassSelberg_slab_of_ne268 below · cited by 1 · depth 28 - Two-term Maass–Selberg relation for an off-diagonal pair
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_axis_continuation_mul_conj_eq_twoTerm_slab_of_ne_of_exists_normOneIdeles268 below · cited by 1 · depth 28 - Maass–Selberg relation on a determinant slab, diagonal case
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_pseudoEisenstein_mul_conj_eq_maassSelberg_slab241 below · cited by 2 · depth 28 - Off-diagonal Maass–Selberg relation on a determinant slab
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_pseudoEisenstein_mul_conj_eq_twoTerm_slab_of_exists_ideleNorm_eq_one_ne242 below · cited by 2 · depth 28 - Summability of archimedean parameters at fixed level and type
AutomorphicForm.exists_forall_integrable_and_summable_rpow_neg_archParam_of_isUnitaryChar_of_pairwise_ne20 below · cited by 2 · depth 28 - Integrability of the truncated continuous kernel, summably in the Eisenstein data
AutomorphicForm.exists_forall_integrable_sum_rightConv_axis_continuation_mul_conj_lambdaT_prod_restrict_canonicalTruncationDomain436 below · cited by 1 · depth 28 - Iwasawa unfolding of flat induced matrix coefficients
AutomorphicForm.exists_forall_integral_rightConv_axis_mul_conj_eq_mul_iwasawa_integral_of_flat10 below · cited by 1 · depth 28 - Integrability of the truncated σ-twisted unipotent term
AutomorphicForm.exists_forall_le_integrableOn_setIntegral_mul_finsum_unipotentNormClass_sub_indicator_constantTerm_canonicalTruncationDomain158 below · cited by 1 · depth 28 - Uniform bound on orthonormal level-N induced sections of listed type
AutomorphicForm.exists_forall_le_of_orthonormal_isInducedSection_principalLevel_archCutSubmodule_of_ne_bot4 below · cited by 3 · depth 28 - Open subgroups of GL₂(ℚₚ) contain a congruence subgroup
AutomorphicForm.exists_forall_mem_of_isOpen_of_congruence0 below · cited by 1 · depth 28 - Truncated unipotent contributions along a slot family over K
AutomorphicForm.exists_forall_mem_slotIndex_integrableOn_and_setIntegral_unipotentCell_eq_weighted_moments_self258 below · cited by 1 · depth 28 - Central character bound along the support of a twisted Hecke word
AutomorphicForm.exists_forall_norm_apply_le_mul_prod_of_isSemiLocalFactorization_of_apply_ne_zero11 below · cited by 1 · depth 28 - Bounded truncated twisted GL₂ kernel on Siegel translates
AutomorphicForm.exists_forall_norm_finsum_sub_indicator_highSet_constantTerm_finsum_borel_le_of_subset_centreCutSiegelSet_translates70 below · cited by 3 · depth 28 - Uniform rapid decay of the Iwasawa integral along the unitary axis
AutomorphicForm.exists_forall_norm_iwasawa_integral_axis_add_norm_deriv_le_mul_rpow_neg_archParam_of_isUnitFactorization18 below · cited by 1 · depth 28 - Cuspidal decay of the twisted Borel kernel minus its constant term
AutomorphicForm.exists_forall_norm_twistedBorelKernel_sub_constantTerm_centralScalar_mul_le_inv_adelicHeight_pow59 below · cited by 1 · depth 28 - One twisted hyperbolic class: truncated term equals weighted orbital integrals
AutomorphicForm.exists_forall_setIntegral_finsum_sigmaConjClassOrbit_sub_indicator_constantTerm_eq_setIntegral_tsum_weight_mul_integral_of_isFactorizableTestFn197 below · cited by 2 · depth 28 - Truncated unipotent term as rank-one Tate integrals over K
AutomorphicForm.exists_forall_setIntegral_finsum_unipotentCell_sub_indicator_constantTerm_eq_sum_mul_setIntegral_rankOne_unram205 below · cited by 1 · depth 28 - Increment form of the GL₂ Maass–Selberg relations on a slab
AutomorphicForm.exists_forall_setIntegral_lambdaT_pseudoEisenstein_mul_conj_sub_eq_maassSelberg_sub_and_sub_eq_twoTerm_sub_slab131 below · cited by 1 · depth 28 - Vanishing of hyperbolic terms for non-σ-invariant ξ_L
AutomorphicForm.exists_forall_setIntegral_tsum_weight_mul_integral_eq_zero_of_not_sigmaInvariant_of_isFactorizableTestFn10 below · cited by 2 · depth 28 - Uniform bound for twisted orbital and weighted orbital integrals
AutomorphicForm.exists_forall_sum_lintegral_orbital_add_weightedOrbital_le_of_isSemiLocalFactorization284 below · cited by 1 · depth 28 - Uniform Hilbert–Schmidt bound for right convolution on cusp forms
AutomorphicForm.exists_forall_sum_setIntegral_norm_sq_rightConv_le_of_orthogonal_of_isCuspidalFn_of_isFundamentalDomain_slab83 below · cited by 4 · depth 28 - Block-wise summability of Hilbert–Schmidt norms of R(f)
AutomorphicForm.exists_forall_sum_sqrt_sum_eLpNorm_convOp_sq_le_of_orthonormal_isotypicCuspSubmodule_principal111 below · cited by 1 · depth 28 - Galois-twisted convolution carries isotypic cusp spaces to a single eigensystem
AutomorphicForm.exists_forall_twistedConvOp_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_isBiInvariantUnder_of_isFundamentalDomain_slab31 below · cited by 1 · depth 28 - Galois twist permutes cut isotypic cuspidal blocks injectively
AutomorphicForm.exists_injOn_forall_twistedConvOp_mem_isotypicCuspSubmodule_comp_unitsMap_inf_archCutSubmodule_of_isFundamentalDomain_slab31 below · cited by 2 · depth 28 - Fujisaki compactness for twisted centralizers of second-kind GL₂ classes
AutomorphicForm.exists_isCompact_forall_exists_mem_sigmaCentralizer_eq_mul_of_ideleNorm_det_eq_one_of_forall_ne_scalar_of_finrank_eq_two7 below · cited by 1 · depth 28 - Local Sobolev bound on adelic GL₂ via archimedean derivative words
AutomorphicForm.exists_isCompact_forall_norm_le_mul_of_forall_eLpNorm_foldr_archDeriv_le19 below · cited by 1 · depth 28 - Twisted centralizer compact modulo Kᵥ-scalars in the non-split case
AutomorphicForm.exists_isCompact_forall_twistedCentralizer_eq_scalar_mul_of_not_isSigmaConjugate_scalar_of_finrank_eq_two1 below · cited by 6 · depth 28 - Properness of the twisted orbit map on the adelic centralizer of a regular semisimple norm
AutomorphicForm.exists_isCompact_setOf_mem_centralizer_normString_twistedConj_mem_subset_twistedCentralizer_mul4 below · cited by 3 · depth 28 - Twisted centraliser of the scalars in adelic GL₂
AutomorphicForm.exists_isCompact_subset_center_forall_eq_mul_mul_scalar_of_inv_mul_sigmaAdelicAct_mem_center51 below · cited by 1 · depth 28 - Complex-place Casimir operators pass onto the test function
AutomorphicForm.exists_isFactorizableTestFn_isBiInvariantUnder_forall_archCasimirAtComplex_convOp_eq_convOp_of_isComplex4 below · cited by 1 · depth 28 - Casimir at a real place passes onto the test function
AutomorphicForm.exists_isFactorizableTestFn_isBiInvariantUnder_forall_archCasimirAt_convOp_eq_convOp_of_isReal4 below · cited by 1 · depth 28 - Nonzero convolution against a factorizable test function at level N
AutomorphicForm.exists_isFactorizableTestFn_rightConv_ne_zero_of_principalLevel_invariant1 below · cited by 1 · depth 28 - Local twisted orbital integral at a central norm, degree two
AutomorphicForm.exists_isHaarMeasure_and_isTwistedOrbitalIntegral_eq_mul_apply_scalar_of_normString_eq_toTensorGL_scalar_of_finrank_eq_two111 below · cited by 1 · depth 28 - Matching archimedean Haar measures on centralizer and twisted centralizer
AutomorphicForm.exists_isHaarMeasure_map_eq_smul_withDensity_arch_of_isNormConjugator_scalar_of_finrank_eq_two2 below · cited by 2 · depth 28 - Spherical flat approximate identity at principal level
AutomorphicForm.exists_isLevelSphericalOfType_principal_flat_tendsto_rightConv_of_finiteDimensional23 below · cited by 2 · depth 28 - Local transfer near a central norm at a finite place
AutomorphicForm.exists_isLocalTestFn_areMatching_nhds_and_central_transfer_of_isNormOf_scalar_of_prime22 below · cited by 1 · depth 28 - Section functions for regular semisimple adelic GL₂ orbital integrals
AutomorphicForm.exists_isSectionFnOn_adeleRing_of_isRegularSemisimple3 below · cited by 8 · depth 28 - Euler factorisation of twisted orbital integrals over places of K
AutomorphicForm.exists_isTwistedOrbitalIntegralOn_baseChange_eq_mul_prod_of_isSemiLocalFactorization_of_isMulRightInvariant6 below · cited by 2 · depth 28 - Existence of twisted orbital integrals at regular semisimple norms
AutomorphicForm.exists_isTwistedOrbitalIntegral_of_isRegularSemisimple_normString_of_isSemiLocalTestFn4 below · cited by 3 · depth 28 - Uniform finite volume bound on determinant slabs for σ-fixed points
AutomorphicForm.exists_lt_top_forall_measure_preimage_le_of_isHaarMeasure_eqLocus_sigmaAdelicAct_id22 below · cited by 1 · depth 28 - Moderate growth of the continued Eisenstein constant term
AutomorphicForm.exists_norm_constantTerm_axis_continuation_le_mul_adelicHeight_rpow_of_mem_of_mem_canonicalTruncationDomain33 below · cited by 3 · depth 28 - Rapid decay of the truncated Eisenstein series on Φ₀
AutomorphicForm.exists_norm_lambdaT_axis_continuation_le_mul_adelicHeight_rpow_neg_of_mem_of_mem_canonicalTruncationDomain126 below · cited by 3 · depth 28 - Truncated hyperbolic σ-class term as weighted twisted orbital integrals
AutomorphicForm.exists_pos_forall_integrable_and_setIntegral_tsum_weight_mul_integral_eq_mul_orbital_add_weightedOrbital_of_isFactorizableTestFn98 below · cited by 1 · depth 28 - Height floor on compact translates of a centre-cut Siegel set
AutomorphicForm.exists_pos_forall_le_adelicHeight_mul_of_mem_centreCutSiegelSet_of_isCompact0 below · cited by 4 · depth 28 - Vanishing of the constant-term defect pairing on the cusp region
AutomorphicForm.exists_pos_forall_setIntegral_sub_constantTerm_mul_eq_zero_canonicalTruncationDomain_inter_lt_adelicHeight38 below · cited by 1 · depth 28 - Affine truncated slab integral over the twisted diagonal centraliser
AutomorphicForm.exists_pos_isFundamentalDomain_forall_setIntegral_indicator_slab_bracket_eq_mul_of_sigmaCentraliser39 below · cited by 7 · depth 28 - A closed twisted diagonal subgroup of GL₂(A_L) carrying Haar measure
AutomorphicForm.exists_subgroup_isClosed_and_mem_iff_diagonal_and_sigmaAdelicAct_mul_inv_mem_center_and_exists_isHaarMeasure0 below · cited by 5 · depth 28 - Flat K-finite induced sections as L^S times Godement sections
AutomorphicForm.exists_sum_mul_godementSection_eq_partialEulerProduct_mul_of_flat_family25 below · cited by 1 · depth 28 - Unipotent difference translate with unit-shell support at p
AutomorphicForm.exists_unipotent_shellSupport_of_shapedRaw_bundle_transl_rat0 below · cited by 1 · depth 28 - Finite-dimensionality of isotypic cusp spaces over a determinant slab
AutomorphicForm.finiteDimensional_isotypicCuspSubmodule_inf_archCutSubmodule_of_isFundamentalDomain337 below · cited by 1 · depth 28 - Parity of the places where δ₀ is not σ-conjugate to a scalar
AutomorphicForm.finite_and_even_ncard_places_not_isSigmaConjugate_scalar_of_finrank_eq_two271 below · cited by 2 · depth 28 - Finiteness of hyperbolic σ-twisted classes meeting a compact support
AutomorphicForm.finite_sep_exists_apply_inv_mul_globalPoints_mul_sigmaAdelicAct_ne_zero_of_diagonal_of_hasCompactSupport5 below · cited by 2 · depth 28 - Derivative words of Casimir-eigen cusp forms bounded on a slab
AutomorphicForm.forall_continuous_isArchSmoothAt_bounded_foldr_archDeriv_of_mem_isotypicCuspSubmodule_principal_of_archCasimir_eq_smul344 below · cited by 2 · depth 28 - Bi-automorphic kernels vanishing a.e. on Φ×Φ vanish
AutomorphicForm.forall_eq_zero_of_ae_prod_restrict_eq_zero_of_apply_globalPoints_mul_of_apply_centralScalar_mul_of_isFundamentalDomain_slab19 below · cited by 2 · depth 28 - Truncated twisted cuspidal kernel integrates blockwise over a fundamental domain
AutomorphicForm.forall_integrableOn_and_setIntegral_lambdaT_tsum_finsum_twistedConvOp_mul_conj_eq_tsum_finsum_setIntegral_of_orthonormal_of_isFundamentalDomain_slab382 below · cited by 2 · depth 28 - Integrability of the central fold of a truncated twisted kernel
AutomorphicForm.integrableOn_mul_finsum_sub_indicator_highSet_constantTerm_finsum_of_hasCompactSupport15 below · cited by 5 · depth 28 - Eisenstein kernel: integrability, joint continuity, automorphy
AutomorphicForm.integrable_and_summable_and_continuous_uncurry_tsum_integral_sum_rightConv_axis_continuation_mul_conj440 below · cited by 2 · depth 28 - Integrability of a bounded truncated twisted GL₂ kernel over Φ×Ω
AutomorphicForm.integrable_mul_finsum_sub_indicator_highSet_constantTerm_finsum_prod_of_forall_norm_le12 below · cited by 3 · depth 28 - Compact (t,t⁻¹)-torus and its table map into X
AutomorphicForm.isCompact_and_exists_torusEmb_and_exists_tableMap_apply_eq_of_sq_eq0 below · cited by 1 · depth 28 - Right convolution preserves cuspidality, smoothness and Hecke eigenvalues
AutomorphicForm.isCuspidalFn_isKfSmooth_levelInvariant_isHeckeCosetEigenfunctionAt_rightConv_of_isFactorizableTestFn_of_support_subset_principal2 below · cited by 1 · depth 28 - Right convolution preserves isotypic cusp forms, principal level
AutomorphicForm.isIsotypicCuspFormAt_rightConv_of_isBiInvariantUnder_principalLevel_of_isFundamentalDomain_slab19 below · cited by 3 · depth 28 - Galois twist of isotypic cusp forms, with twisted central character
AutomorphicForm.isIsotypicCuspFormAt_sigmaSectionActOn_comp_unitsMap_of_isFundamentalDomain_slab14 below · cited by 3 · depth 28 - Unimodularity of the σ-fixed subgroup of GL₂(A_L)
AutomorphicForm.isMulRightInvariant_of_isHaarMeasure_eqLocus_sigmaAdelicAct_id4 below · cited by 2 · depth 28 - Unimodularity of GL₂(K_∞): Haar measures are right invariant
AutomorphicForm.isMulRightInvariant_of_isHaarMeasure_generalLinearGroup_infiniteAdeleRing4 below · cited by 2 · depth 28 - Unimodularity of GL₂(A) for semilocal A
AutomorphicForm.isMulRightInvariant_of_isHaarMeasure_generalLinearGroup_of_finite_maximalSpectrum0 below · cited by 4 · depth 28 - Rapid decay on Siegel sets of a smoothed cusp vector over ℚ
AutomorphicForm.isRapidlyDecreasingOnSiegelSets_rightConv_of_isCuspAutomorphicFnAt_of_norm_apply_eq_one_rat74 below · cited by 1 · depth 28 - Semi-local factorisation over K/K of Hecke-word test functions
AutomorphicForm.isSemiLocalFactorization_self_of_finComponent_factorization1 below · cited by 4 · depth 28 - Same central norm string implies σ-conjugacy for GL₂ over adeles
AutomorphicForm.isSigmaConjugate_adeleRing_of_normString_eq_of_normString_mem_center_of_finrank_eq_two4 below · cited by 1 · depth 28 - Pairings of invariant functions over determinant slabs and the σ-twist
AutomorphicForm.memLp_and_setIntegral_mul_conj_eq_and_setIntegral_sigmaSectionActOn_eq_of_isFundamentalDomain_slab8 below · cited by 2 · depth 28 - L²-continuity of truncated continued Eisenstein families
AutomorphicForm.memLp_two_lambdaT_and_tendsto_eLpNorm_lambdaT_sub_restrict_canonicalTruncationDomain_of_rapidlyDecreasing_family33 below · cited by 3 · depth 28 - Determinant-norm twisting of cuspidal classes and cut traces
AutomorphicForm.mem_cuspClasses_iff_twist_mem_cuspClasses_and_cutTrace_eq_cutTrace_twist_mul_ideleNorm_det_rpow_of_subset_slab13 below · cited by 1 · depth 28 - Covolume rates of GL₂(A_K) and a twisted centraliser
AutomorphicForm.mul_eq_two_mul_of_forall_isFundamentalDomain_twistedCentralizer_measure_inter_ideleNorm_det_Icc_of_forall_ne_scalar_of_finrank_eq_two489 below · cited by 1 · depth 28 - Norm string of a central idele equals its idelic norm
AutomorphicForm.normString_map_baseChangeEquiv_symm_centralScalar_eq_toTensorGL_centralScalar_idelicNorm3 below · cited by 3 · depth 28 - Rapid decay of continued Eisenstein series minus its constant term
AutomorphicForm.norm_sub_constantTerm_le_mul_rpow_neg_of_axis_continuation_family110 below · cited by 3 · depth 28 - Automorphy of the pseudo-Eisenstein series on GL₂
AutomorphicForm.pseudoEisenstein_globalPoints_mul_eq_of_forall_mem_borelSubgroup_of_summable0 below · cited by 9 · depth 28 - Commutation of two right convolutions at principal level
AutomorphicForm.rightConv_rightConv_comm_of_isLevelSphericalOfType_principal1 below · cited by 1 · depth 28 - Associativity of right convolution on GL₂(A_K)
AutomorphicForm.rightConv_rightConv_inv_eq_rightConv_rightConv0 below · cited by 4 · depth 28 - Truncated twisted hyperbolic term as a finite sum over Δ_φ
AutomorphicForm.setIntegral_mul_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_sum_of_hasCompactSupport21 below · cited by 2 · depth 28 - Vanishing of twisted convolution pairing for non-σ-invariant central character
AutomorphicForm.setIntegral_twistedConvOp_mul_conj_eq_zero_of_exists_apply_act_ne_of_isFundamentalDomain_slab11 below · cited by 3 · depth 28 - Twisted hyperbolic cell at σ=1 equals untwisted cell
AutomorphicForm.setIntegral_twistedHyperbolicCell_self_one_eq_setIntegral_hyperbolicCell0 below · cited by 3 · depth 28 - Trivial twist: σ=1 unipotent cell is untwisted
AutomorphicForm.setIntegral_twistedUnipotentCell_self_one_eq_setIntegral_unipotentCell0 below · cited by 1 · depth 28 - Vanishing of the unipotent fold against a character ramified on T
AutomorphicForm.setIntegral_unipotentCell_fold_eq_zero_of_exists_localUnit_apply_ne_one4 below · cited by 1 · depth 28 - Unipotent difference translate preserves the shaped bundle at p
AutomorphicForm.shapedRaw_bundle_sub_translate_unipotent_transl_rat106 below · cited by 1 · depth 28 - Raw Whittaker bundle over ℚ and unramified laws
AutomorphicForm.shapedRaw_rawBundle_transl_rat98 below · cited by 1 · depth 28 - Slot-family assembly of the explicit unipotent moments
AutomorphicForm.sum_slotFamilyCoeff_mul_unipotentMoments_eq_mul_sum_laurentCoeff_add_sum_laurentCoeff_edge2 below · cited by 1 · depth 28 - Archimedean twisted orbital integral at a central norm class
AutomorphicForm.twistedOrbitalIntegral_eq_neg_one_pow_mul_orbitalIntegral_scalar_arch_of_finrank_eq_two122 below · cited by 1 · depth 28 - Holomorphy and joint continuity of the intertwining integral for Re s > 1/2
AutomorphicForm.analyticOnNhd_and_continuousOn_weylIntertwiningIntegral_family_of_re_gt_half81 below · cited by 2 · depth 29 - Unramified descent of an idele class character along the norm
AutomorphicForm.apply_localUnit_eq_one_of_comp_idelicNorm_of_forall_apply_localUnit_eq_one_of_ramificationIdx_eq_one7 below · cited by 1 · depth 29 - Ascent of unramifiedness along the idelic norm
AutomorphicForm.apply_localUnit_eq_one_of_eq_comp_idelicNorm_of_forall_apply_localUnit_under_eq_one_of_ramificationIdx_eq_one7 below · cited by 1 · depth 29 - Central local unit invariance of Hecke-word test functions
AutomorphicForm.apply_mul_centralScalar_localUnit_eq_of_glArch_mul_glFin_heckeWord_of_not_mem0 below · cited by 1 · depth 29 - Central local-unit invariance of a semi-locally factorised test function
AutomorphicForm.apply_mul_centralScalar_localUnit_eq_of_isSemiLocalFactorization_heckeWord_of_under_not_mem0 below · cited by 1 · depth 29 - Uniformiser scalars act by the raw central value bᵥ/cNorm(v)
AutomorphicForm.apply_mul_placeEmbed_scalarPi_eq_toRawCentral_b_mul_of_isIsotypicCuspFormAt0 below · cited by 1 · depth 29 - Vanishing at 1 from shrinking elliptic orbital integrals
AutomorphicForm.apply_one_eq_zero_of_isLocallyConstant_of_forall_exists_integral_integral_eq_zero0 below · cited by 1 · depth 29 - Archimedean central transfer from one-place central comparisons
AutomorphicForm.areMatchingArch_central_transfer_of_scalar_of_forall_conjAe_of_forall_algHom38 below · cited by 1 · depth 29 - Central character μν of the continued Eisenstein series
AutomorphicForm.axis_continuation_bruhatEisenstein_centralScalar_mul_eq_of_isArchKFinite_family0 below · cited by 11 · depth 29 - Continuity of the archimedean flows in GL₂(A_K)
AutomorphicForm.continuous_archFlowAt_and_continuous_archFlowAtComplex0 below · cited by 4 · depth 29 - Continuity of the GL₂ pseudo-Eisenstein series for Re s>1/2
AutomorphicForm.continuous_pseudoEisenstein_of_isInducedSection_of_re_gt_half9 below · cited by 3 · depth 29 - Translation differences preserve automorphic shape data
AutomorphicForm.continuous_rapidlyDecreasing_whittakerCoefficient_sub_translate0 below · cited by 1 · depth 29 - Continuity of the Weyl intertwining integral for Re s > 1/2
AutomorphicForm.continuous_weylIntertwiningIntegral_of_re_gt_half10 below · cited by 6 · depth 29 - Twisted commutant acts simply transitively on L²
AutomorphicForm.existsUnique_mul_eq_mul_map_and_mulVec_eq_of_forall_ne_scalar_of_finrank_eq_two1 below · cited by 9 · depth 29 - Completed normalised intertwining operator across the axis
AutomorphicForm.exists_analyticOnNhd_normalizedIntertwining_completedL_mul_axis_continuation_weylIntertwiningIntegral_eq_mul_of_flat37 below · cited by 2 · depth 29 - Twisted Eisenstein term: slope, Eisenstein-table atoms, atom-free remainder
AutomorphicForm.exists_atomic_forall_tendsto_of_eq_mul_tsum_integral_sum_rightConv_mul_setIntegral_lambdaT_mul_conj_lambdaT_sigmaAdelicAct_of_isSemiLocalFactorization494 below · cited by 1 · depth 29 - Uniform bounds, parameters and summability for GL(2) Eisenstein data
AutomorphicForm.exists_bound_card_and_archParam_weight_and_summable_of_orthonormal_flat_isInducedSection_family_ed240 below · cited by 6 · depth 29 - A uniform transfer constant for hyperbolic intercepts
AutomorphicForm.exists_const_forall_exists_windingDatum_hyperbolicIntercept_sub_finrank_mul_const_mul_sum_eq_sum_satakeLaurent_mul_coeff_of_eq_affine1,456 below · cited by 1 · depth 29 - Entire normalisation of the non-constant term of an Eisenstein family
AutomorphicForm.exists_entire_eq_mul_bruhatEisenstein_sub_constantTerm_norm_le_rpow_neg_of_isArchKFinite_family_of_unitary108 below · cited by 1 · depth 29 - Entire continuation of normalised Whittaker coefficients of flat Eisenstein families
AutomorphicForm.exists_entire_whittakerCoefficient_diagOne_continuation_of_flat_family_of_unitary76 below · cited by 1 · depth 29 - Standing data for the split hyperbolic family over K
AutomorphicForm.exists_eq_archHaarK_torusFamily_isOrbitalIntegral_centralScalar_mul_diagUnits2_of_isArchTestFactor_of_isLocalTestFn13 below · cited by 2 · depth 29 - Flat family through a K-finite induced section
AutomorphicForm.exists_family_forall_isInducedSection_and_eq_of_isArchKFinite_of_isKfSmooth4 below · cited by 2 · depth 29 - Finiteness of split rational classes meeting a compact set
AutomorphicForm.exists_finset_forall_apply_conj_centralScalar_mul_diagUnits2_eq_zero_of_hasCompactSupport1 below · cited by 3 · depth 29 - Local twisted section functions from a global one
AutomorphicForm.exists_finset_forall_isTwistedSectionFnOn_indicator_semiLocalIntegralSet_of_isTwistedSectionFnOn_baseChange_of_isMulRightInvariant0 below · cited by 1 · depth 29 - Finite-adelic Godement sections realise K_f-smooth Borel-equivariant functions
AutomorphicForm.exists_finset_sum_mul_prod_localZeta_bottomRow_eq_of_isKfSmooth3 below · cited by 1 · depth 29 - Uniform weight window at complex places for type sums
AutomorphicForm.exists_forall_abs_le_of_apply_mul_archCircleAt_eq_zpow_mul_of_mem_iSup_archTypeSubmoduleAt5 below · cited by 2 · depth 29 - Flow-chart derivatives dominated by words in archimedean derivations
AutomorphicForm.exists_forall_contDiff_norm_iteratedFDeriv_comp_flowChart_le_sum_foldr_archDeriv7 below · cited by 1 · depth 29 - Single-letter L² bound at a complex place for Casimir eigenfunctions
AutomorphicForm.exists_forall_eLpNorm_archDerivAtComplex_foldr_le_mul_sqrt_mul_eLpNorm_of_mem_archCutSubmodule_of_archCasimir_eq_smul46 below · cited by 1 · depth 29 - Single-letter L² derivative bound at a real place
AutomorphicForm.exists_forall_eLpNorm_archDerivAt_foldr_le_mul_sqrt_mul_eLpNorm_of_mem_archCutSubmodule_of_archCasimir_eq_smul41 below · cited by 1 · depth 29 - Polynomial counting bound for archimedean parameters of idele class characters
AutomorphicForm.exists_forall_finite_and_ncard_archParam_spread_le_of_isUnitaryChar_of_pairwise_ne_normOneIdeles17 below · cited by 2 · depth 29 - Slope transfer for the twisted hyperbolic term
AutomorphicForm.exists_forall_hyperbolicSlope_eq_mul_sum_slotFamilyCoeff_mul_hyperbolicSlope_of_eq_affine1,015 below · cited by 1 · depth 29 - Slab Maass–Selberg relation in the range Re s<Re s'
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_pseudoEisenstein_mul_conj_eq_maassSelberg_slab_of_re_lt_re100 below · cited by 1 · depth 29 - Off-diagonal Maass–Selberg relation on a determinant slab
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_pseudoEisenstein_mul_conj_eq_twoTerm_slab_of_exists_ideleNorm_eq_one_ne_of_re_lt_re101 below · cited by 1 · depth 29 - Cusp cancellation for the truncated twisted class sum
AutomorphicForm.exists_forall_integrableOn_tsum_indicator_highSet_mul_twistedOrbital_sub_indicator_mul_tsum_integral_unipotentGL2_and_setIntegral_eq_zero_of_isFactorizableTestFn115 below · cited by 1 · depth 29 - Unipotent term in Iwasawa coordinates via rank-one Tate integrals
AutomorphicForm.exists_forall_integral_iwasawa_cuspKernel_sub_cuspTruncation_eq_sum_mul_setIntegral_rankOne_of_sigmaInvariant_unram_ed2197 below · cited by 1 · depth 29 - Haar comparison for flow-chart boxes in GL₂(A_K)
AutomorphicForm.exists_forall_lintegral_comp_flowChart_le_mul_lintegral_of_forall_mul_eq9 below · cited by 1 · depth 29 - Uniform L² bound for the axis derivative of R(s)
AutomorphicForm.exists_forall_lintegral_norm_sq_deriv_normalizedIntertwining_axis_le_of_completedL_mul_weylIntertwiningIntegral_eq_of_flat140 below · cited by 1 · depth 29 - Uniform axis L²(K) bound for the normalised intertwining operator
AutomorphicForm.exists_forall_lintegral_norm_sq_normalizedIntertwining_axis_le_of_completedL_mul_weylIntertwiningIntegral_eq_of_flat319 below · cited by 1 · depth 29 - Uniform moderate growth of flat Eisenstein series on the truncation domain
AutomorphicForm.exists_forall_norm_axis_continuation_le_mul_pow_archParam_weight_mul_adelicHeight_rpow_of_mem_canonicalTruncationDomain_of_flat409 below · cited by 2 · depth 29 - Uniform polynomial growth of unitary GL₂ Eisenstein series
AutomorphicForm.exists_forall_norm_axis_continuation_le_mul_pow_archParam_weight_of_isCompact_of_flat413 below · cited by 11 · depth 29 - Boundedness of χ∘det on a determinant slab in Siegel sets
AutomorphicForm.exists_forall_norm_chiDet_le_of_mem_setOf_ideleNorm_det_inter_iUnion_image_centreCutSiegelSet4 below · cited by 1 · depth 29 - Rapid cuspidal decay of twisted GL₂ kernel minus constant term
AutomorphicForm.exists_forall_norm_finsum_borel_div_mem_sub_constantTerm_centralScalar_mul_le_inv_adelicHeight_pow59 below · cited by 2 · depth 29 - Induced sections on GL₂(A_F) grow like H^{σ+1/2}
AutomorphicForm.exists_forall_norm_le_mul_adelicHeight_rpow_of_isInducedSection6 below · cited by 4 · depth 29 - Rapid decay of axis matrix coefficients for factorizable test functions
AutomorphicForm.exists_forall_norm_rightConv_axis_pairing_add_norm_deriv_le_mul_rpow_neg_archParam_of_isFactorizableTestFn28 below · cited by 6 · depth 29 - Uniform Poisson tail bound for unipotent slices of a test function
AutomorphicForm.exists_forall_norm_tsum_sub_inv_measure_mul_integral_comp_unipotentGL2_le_of_isCompact46 below · cited by 2 · depth 29 - Integrated spectral expansion of the truncated σ-twisted continuous kernel
AutomorphicForm.exists_forall_setIntegral_lambdaT_sigmaAdelicAct_sub_twistedConvOp_sub_chiDet_eq_mul_tsum_integral_sum_rightConv_mul_setIntegral_lambdaT_mul_conj_lambdaT_sigmaAdelicAct1,291 below · cited by 1 · depth 29 - Rectangle form of the GL₂ spectral kernel expansion
AutomorphicForm.exists_forall_setIntegral_prod_restrict_canonicalTruncationDomain_finsum_integral_centralScalar_sub_tsum_convOp_sub_finsum_chiDet_eq_mul_setIntegral_tsum_integral_sum_rightConv_axis_continuation1,234 below · cited by 1 · depth 29 - Bound for twisted hyperbolic orbital sums of semi-local translates
AutomorphicForm.exists_forall_sum_lintegral_orbital_add_weightedOrbital_le_mul_prod_card_of_isSemiLocalFactorization_translates282 below · cited by 1 · depth 29 - Base change of the twisted commutant of δ₀⊗ 1
AutomorphicForm.exists_homeomorph_twistedCommutant_map_mul_scalar_forall_coe_eq_sum_map_tmul_of_linearIndependent0 below · cited by 10 · depth 29 - Properness of the centre of GL₂(A_K)
AutomorphicForm.exists_isCompact_forall_mem_of_inv_mul_globalPoints_mul_centralScalar_mul_mem_of_isCompact0 below · cited by 3 · depth 29 - Properness of the twisted orbit map modulo the twisted centraliser
AutomorphicForm.exists_isCompact_setOf_mem_centralizer_normString_twistedConj_mem_subset_twistedCentralizer_mul_of_isArtinianRing1 below · cited by 2 · depth 29 - Coupled measures from Gram normalisation at positive scalars
AutomorphicForm.exists_isNormConjugator_and_coupled_of_gram_conjAe_of_pos0 below · cited by 2 · depth 29 - Norm-conjugator coupling Haar measures when δ is σ-conjugate to a scalar
AutomorphicForm.exists_isNormConjugator_and_coupled_smul_of_isSigmaConjugate_scalar_of_finrank_eq_two0 below · cited by 1 · depth 29 - σ-conjugacy to a scalar versus norms from L⊗_KF
AutomorphicForm.exists_isSigmaConjugate_scalar_iff_algebraMap_mem_range_norm_of_finrank_eq_two0 below · cited by 1 · depth 29 - Gram-normalised measure on M₂(L⊗_K K_∞) descends to M₂(K_∞)
AutomorphicForm.exists_map_eq_smul_withDensity_gram_infiniteAdeleRing_of_map_includeRight_eq_smul_withDensity_gram_of_finrank_eq_two0 below · cited by 1 · depth 29 - Local transfer near a singular norm, with central identities
AutomorphicForm.exists_nhds_forall_exists_areMatchingLocal_and_central_of_not_isRegularSemisimple_normString_of_prime21 below · cited by 1 · depth 29 - Haar measure on the adelic diagonal torus via diag(p₁p₂,p₁)
AutomorphicForm.exists_pos_forall_integral_subgroup_eq_mul_integral_prod_centralScalar_mul_diagUnits2_one2 below · cited by 2 · depth 29 - Truncation domain high in the cusp: Borel fundamental domain
AutomorphicForm.exists_pos_forall_isFundamentalDomain_borelSubgroup_canonicalTruncationDomain_inter_lt_adelicHeight22 below · cited by 1 · depth 29 - Height floor on the canonical truncation domain
AutomorphicForm.exists_pos_forall_le_adelicHeight_and_adelicHeight_globalPoints_mul_le_inv_of_mem_canonicalTruncationDomain19 below · cited by 13 · depth 29 - Haar measure on the twisted diagonal centraliser in GL₂(A_L)
AutomorphicForm.exists_pos_forall_lintegral_sigmaCentraliser_eq_mul_lintegral_lintegral_centralScalar_mul_diagOne25 below · cited by 3 · depth 29 - Idèle norm of det g on the support of Hecke words
AutomorphicForm.exists_pos_forall_mul_prod_le_ideleNorm_det_le_of_isSemiLocalFactorization_of_apply_ne_zero2 below · cited by 1 · depth 29 - Euler factorisation of the Godement section on the maximal compact
AutomorphicForm.exists_pos_godementSection_mul_tprod_eq_mul_prod_localZeta_of_mem_adelicMaximalCompact4 below · cited by 1 · depth 29 - Explicit Hecke coset relation at v for any uniformiser
AutomorphicForm.exists_sum_apply_mul_placeEmbed_repSome_add_apply_mul_placeEmbed_repInf_eq_of_isHeckeCosetEigenfunctionAt3 below · cited by 1 · depth 29 - Archimedean Godement sections realise K_∞-finite functions on GL₂
AutomorphicForm.exists_sum_mul_prod_localZeta_bottomRow_eq_of_isArchKFinite12 below · cited by 1 · depth 29 - One winding datum for all K-side Hecke words
AutomorphicForm.exists_windingDatum_forall_heckeWord_mul_sum_slotFamilyCoeff_mul_sum_classIntegral_eq_sum_satakeLaurent_mul_coeff116 below · cited by 1 · depth 29 - Finiteness and convergence of twisted orbital integrands along coset representatives
AutomorphicForm.finite_setOf_exists_apply_twistedOrbitalIntegrand_ne_zero_and_tsum_lintegral_lt_top9 below · cited by 1 · depth 29 - Finiteness of rational classes mod centre meeting a compact set
AutomorphicForm.finite_setOf_exists_mem_exists_inv_mul_globalPoints_out_mul_centralScalar_mul_mem_of_isCompact1 below · cited by 4 · depth 29 - Words in archimedean derivations preserve level invariance and central characters
AutomorphicForm.foldr_archDeriv_mul_eq_of_forall_mul_eq0 below · cited by 1 · depth 29 - Uniform rapid decay of truncated unitary Eisenstein series
AutomorphicForm.forall_exists_forall_norm_lambdaT_axis_continuation_le_mul_pow_archParam_weight_mul_adelicHeight_rpow_neg_of_mem_canonicalTruncationDomain_of_flat277 below · cited by 1 · depth 29 - Haar measure on centralisers of regular diagonal elements
AutomorphicForm.forall_exists_isHaarMeasure_centralizer_globalPoints_integral_eq_mul_integral_prod_diagUnits21 below · cited by 5 · depth 29 - Integrability of the truncated unipotent-type fold over the centre
AutomorphicForm.forall_integrableOn_finsum_unipotentCell_sub_indicator_constantTerm_fold_unram30 below · cited by 1 · depth 29 - Class-block summable majorant for the cuspidal kernel
AutomorphicForm.forall_isCompact_exists_summable_forall_finsum_norm_convOp_mul_conj_le_of_orthonormal_isotypicCuspSubmodule503 below · cited by 2 · depth 29 - Locally uniform absolute convergence of the twisted cuspidal kernel
AutomorphicForm.forall_isCompact_exists_tsum_norm_finsum_twistedConvOp_mul_conj_le_of_orthonormal_of_isFundamentalDomain_slab56 below · cited by 1 · depth 29 - Unfolding one twisted hyperbolic class into orbital integrals
AutomorphicForm.integrableOn_finsum_sigmaConjClassOrbit_and_setIntegral_eq_tsum_integral_of_leftCosetRepresentatives2 below · cited by 1 · depth 29 - Unfolding one σ-twisted hyperbolic class over the centraliser quotient
AutomorphicForm.integrableOn_tsum_bracket_mul_twistedOrbital_and_setIntegral_eq_mul_integral_setIntegral_indicator_bracket_mul18 below · cited by 2 · depth 29 - Class-by-class trace-norm summability of twisted convolution kernels
AutomorphicForm.integrableOn_twistedConvOp_mul_conj_and_summable_setIntegral_norm_finsum_twistedConvOp_mul_conj_of_orthonormal_of_isFundamentalDomain_slab367 below · cited by 1 · depth 29 - Integrability of (1+sumᵥ(|t+τᵥ|+|t-τ'ᵥ|))^{-B} for B≥ 2
AutomorphicForm.integrable_one_add_sum_abs_add_abs_sub_rpow_neg0 below · cited by 1 · depth 29 - Convergence of hyperbolic twisted orbital integrals over HbackslashGL₂(mathbb A_L)
AutomorphicForm.integrable_twistedOrbital_and_weighted_and_exists_height_mul_le_of_diagonal_of_norm_ne_one53 below · cited by 3 · depth 29 - Vanishing orbital integrals kill the affine-chart integral
AutomorphicForm.integral_conj_affineChart_eq_zero_of_forall_isOrbitalIntegral_eq_zero3 below · cited by 1 · depth 29 - Twisted orbital integral transforms by a central character value
AutomorphicForm.integral_mul_apply_inv_mul_mul_sigmaAdelicAct_centralScalar_mul_eq_of_inv_mul_mul_sigmaAdelicAct_eq_mul_centralScalar0 below · cited by 5 · depth 29 - Hecke words multiply η∘det integrals by local character sums
AutomorphicForm.integral_mul_chiDet_eq_prod_sum_localChar_mul_integral_of_isSemiLocalFactorization0 below · cited by 1 · depth 29 - Iwasawa evaluation of a truncated pairing of induced sections
AutomorphicForm.integral_rationalTorusUnipotentQuotient_section_mul_conj_eq_mul_setIntegral_iwasawa24 below · cited by 3 · depth 29 - Right translates of cusp-automorphic functions over ℚ
AutomorphicForm.isCuspAutomorphicFnAt_comp_mul_right_and_sub_of_rightConv_eq_rat102 below · cited by 1 · depth 29 - Slab-cut truncated shell integrals on a twisted diagonal centraliser
AutomorphicForm.isFundamentalDomain_image_and_forall_setLIntegral_indicator_slab_bracket_eq_of_lintegral_eq30 below · cited by 2 · depth 29 - Right convolution preserves Hecke eigenfunctions at good places
AutomorphicForm.isHeckeCosetEigenfunctionAt_rightConv_of_isBiInvariantUnder_principalLevel_of_not_dvd8 below · cited by 1 · depth 29 - Weyl intertwining integral of an induced section, Re s>1/2
AutomorphicForm.isInducedSection_and_continuous_weylIntertwiningIntegral_of_re_gt_half12 below · cited by 6 · depth 29 - Right convolution preserves isotypic cusp forms at level N
AutomorphicForm.isIsotypicCuspFormAt_rightConv_of_isBiInvariantUnder_of_isFundamentalDomain_slab19 below · cited by 3 · depth 29 - Galois twist of an isotypic cusp form on GL₂
AutomorphicForm.isIsotypicCuspFormAt_sigmaSectionActOn_of_isFundamentalDomain_slab14 below · cited by 1 · depth 29 - Twisting an isotypic cusp form by ‖det‖^{-w/2}
AutomorphicForm.isIsotypicCuspFormAt_twist_mul_ideleNorm_det_rpow_of_subset_slab10 below · cited by 1 · depth 29 - Local factors of a right-invariant adelic twisted-centralizer measure
AutomorphicForm.isMulRightInvariant_arch_and_place_of_isMulRightInvariant_of_integral_twistedCentralizer_eq_mul_prod0 below · cited by 1 · depth 29 - Negative central scalar: twisted orbital integral equals minus orbital integral
AutomorphicForm.isOrbitalIntegralOn_scalar_neg_of_isTwistedOrbitalIntegralOn_conjAe_of_gram_of_nhds_forall_isRegularSemisimple_of_neg41 below · cited by 1 · depth 29 - Archimedean central transfer of matching at a positive scalar
AutomorphicForm.isOrbitalIntegralOn_scalar_of_isTwistedOrbitalIntegralOn_conjAe_of_nhds_forall_isRegularSemisimple_of_pos34 below · cited by 2 · depth 29 - Central transfer at a split complex place
AutomorphicForm.isOrbitalIntegralOn_scalar_of_isTwistedOrbitalIntegralOn_of_algHom_complex_of_nhds_forall_isRegularSemisimple22 below · cited by 2 · depth 29 - Central transfer at a split real place
AutomorphicForm.isOrbitalIntegralOn_scalar_of_isTwistedOrbitalIntegralOn_of_algHom_real_of_nhds_forall_isRegularSemisimple26 below · cited by 2 · depth 29 - Hecke word indicators are semi-local test functions
AutomorphicForm.isSemiLocalTestFn_sum_indicator_semiLocalIntegralSet_word0 below · cited by 3 · depth 29 - Right convolution preserves cusp forms and produces smoothness
AutomorphicForm.isSmoothCuspAutomorphicFnAt_rightConv_principalLevel_of_isFundamentalDomain_slab11 below · cited by 1 · depth 29 - Invertibility of nonzero y with δ σ(y)∈ y M₂
AutomorphicForm.isUnit_of_mul_map_sigmaTensor_eq_mul_of_not_isSigmaConjugate_scalar_of_finrank_eq_two0 below · cited by 8 · depth 29 - Level-N invariance forces triviality of μᵥ,νᵥ on congruence units
AutomorphicForm.localChar_eq_one_of_isInducedSection_etaFst_etaSnd_of_ne_zero_of_principalLevel_of_valued_sub_one_le3 below · cited by 1 · depth 29 - Translation on the twisted commutant scales Haar by ‖det t‖_L⁻¹
AutomorphicForm.map_mul_addHaar_twistedCommutant_eq_inv_distribHaarChar_det_smul_of_normString_eq_toTensorGL_centralScalar_of_finrank_eq_two2 below · cited by 1 · depth 29 - L²-boundedness of truncated pseudo-Eisenstein series on a slab
AutomorphicForm.memLp_two_lambdaT_pseudoEisenstein_restrict_canonicalTruncationDomain46 below · cited by 7 · depth 29 - Determinant twists preserve the archimedean type cut
AutomorphicForm.mul_ideleNorm_det_rpow_mem_archCutSubmodule0 below · cited by 1 · depth 29 - Increment of the truncated Petersson pairing across a height shell
AutomorphicForm.peterssonIntegral_lambdaT_sub_eq_integral_constantTerm_mul_conj_constantTerm31 below · cited by 2 · depth 29 - Covolume rate for GL₂ over the adeles
AutomorphicForm.rate_eq_mul_discr_sq_mul_dedekindZeta_two_mul_residue_of_forall_isFundamentalDomain_globalPoints_inter_ideleNorm_det_Icc94 below · cited by 1 · depth 29 - Right convolution commutes with the ‖det‖-twist
AutomorphicForm.rightConv_mul_ideleNorm_det_rpow_neg_half0 below · cited by 1 · depth 29 - Vanishing of the central and elliptic fold against a character
AutomorphicForm.setIntegral_centralEllipticPart_fold_eq_zero_of_forall_apply_mul_centralScalar_eq_of_ne_one1 below · cited by 1 · depth 29 - Vanishing of a twisted determinant-character integral over a fundamental domain
AutomorphicForm.setIntegral_chiDet_sigmaAdelicAct_mul_chiDet_inv_eq_zero_of_isFundamentalDomain_slab9 below · cited by 1 · depth 29 - Vanishing of the hyperbolic ξ-fold for a ramified central character
AutomorphicForm.setIntegral_hyperbolicCell_fold_eq_zero_of_forall_apply_mul_centralScalar_eq_of_ne_one2 below · cited by 1 · depth 29 - Orthogonality of isotypic cusp forms for distinct Hecke eigensystems
AutomorphicForm.setIntegral_mul_conj_eq_zero_of_mem_isotypicCuspSubmodule_principalLevel_of_ne_of_isFundamentalDomain_slab16 below · cited by 2 · depth 29 - Unfolding a truncated hyperbolic constant term over the centre
AutomorphicForm.setIntegral_mul_indicator_highSet_constantTerm_finsum_eq_indicator_mul_tsum_integral_unipotentGL2_twistedOrbital18 below · cited by 2 · depth 29 - Vanishing of the twisted hyperbolic ξ_L-fold over a fundamental domain
AutomorphicForm.setIntegral_twistedHyperbolicCell_fold_eq_zero_of_forall_apply_mul_sigmaAdelicAct_centralScalar_eq_of_ne_one2 below · cited by 1 · depth 29 - Vanishing of the ξ-twisted unipotent fold under central invariance
AutomorphicForm.setIntegral_unipotentCell_fold_eq_zero_of_forall_apply_mul_centralScalar_eq_of_ne_one2 below · cited by 1 · depth 29 - Integrability of one truncated twisted hyperbolic class sum
AutomorphicForm.setLIntegral_tsum_norm_bracket_mul_twistedOrbital_lt_top_and_integrableOn99 below · cited by 1 · depth 29 - Integral twisted orbits at unramified places of GL₂
AutomorphicForm.setOf_mem_centralizer_normString_twistedConj_mem_semiLocalIntegralSet_subset_twistedCentralizer_mul_of_ramificationIdx_eq_one1 below · cited by 1 · depth 29 - Fixed points of σ ⊗ id on L ⊗_K A
AutomorphicForm.sigmaTensor_apply_eq_self_iff0 below · cited by 4 · depth 29 - Unweighted split-class expansion of the ground-field hyperbolic slope
AutomorphicForm.slope_eq_sum_unweighted_classIntegral_diagUnits2_of_inversionClosed_of_hyperbolicTerm_eq_affine227 below · cited by 1 · depth 29 - Polynomial sparsity of spreads gives summable archimedean weights
AutomorphicForm.summable_integral_rpow_neg_and_summable_rpow_neg_of_ncard_spread_le0 below · cited by 1 · depth 29 - Archimedean central base change comparison with Kottwitz sign
AutomorphicForm.twistedOrbitalIntegral_eq_neg_one_pow_mul_orbitalIntegral_scalar_arch_of_forall_conjAe_of_forall_gram_of_forall_algHom95 below · cited by 1 · depth 29 - Twisted orbital integral is minus the scalar orbital integral
AutomorphicForm.twistedOrbitalIntegral_eq_neg_orbitalIntegral_scalar_of_not_isSigmaConjugate_of_finrank_eq_two91 below · cited by 1 · depth 29 - Mass formula for the σ-twisted centralizer of δ
AutomorphicForm.two_mul_rate_eq_mul_discr_sq_mul_dedekindZeta_two_mul_residue_of_forall_isFundamentalDomain_twistedCentralizer_inter_ideleNorm_det_Icc_of_forall_ne_scalar_of_finrank_eq_two444 below · cited by 1 · depth 29 - Fibrewise constancy of symmetric data of an unramified character pair
AutomorphicForm.apply_det_heckeGen_add_eq_and_mul_eq_and_cNorm_eq_of_under_eq_of_sigmaInvariant_or_sigmaReversed7 below · cited by 1 · depth 30 - Harish-Chandra limit formula at a scalar matrix, explicit constant
AutomorphicForm.apply_scalar_eq_const_mul_of_isOrbitalIntegralOn_rotation_nhdsGT_of_tendsto_ellipticTransform7 below · cited by 2 · depth 30 - Vanishing form of the limit formula on GL₂(ℂ)
AutomorphicForm.apply_scalar_eq_zero_of_nhds_forall_isRegularSemisimple_isOrbitalIntegralOn_complex_eq_zero10 below · cited by 1 · depth 30 - Harish-Chandra limit formula on GL₂(ℝ): vanishing form
AutomorphicForm.apply_scalar_eq_zero_of_nhds_forall_isRegularSemisimple_isOrbitalIntegralOn_real_eq_zero14 below · cited by 2 · depth 30 - Complex-place Casimir operators commute with archimedean derivative words
AutomorphicForm.archCasimirAtComplex_and_archCasimirBarAtComplex_foldr_archDeriv_eq_foldr_archDeriv2 below · cited by 1 · depth 30 - Casimir at a real place commutes with derivation words
AutomorphicForm.archCasimirAt_foldr_archDeriv_eq_foldr_archDeriv_archCasimirAt3 below · cited by 1 · depth 30 - Value of the archimedean base-change map on a pure tensor
AutomorphicForm.archIdent_tmul_apply0 below · cited by 4 · depth 30 - Closed form and axis bounds for N(w)=(-i)^kΓ_ℝ-quotient
AutomorphicForm.archIntertwiningReal_eq_prod_div_and_norm_eq_one_and_norm_deriv_le_of_re_eq_zero0 below · cited by 1 · depth 30 - Uniform pure-tensor big-cell expansion of flat induced families
AutomorphicForm.bigCell_eq_sum_pureTensor_of_flat_family_of_restrict_eq9 below · cited by 2 · depth 30 - Big-cell pure-tensor decomposition with archimedean type parity
AutomorphicForm.bigCell_eq_sum_pureTensor_of_flat_family_of_type_parity6 below · cited by 4 · depth 30 - Rapid decay of the Bruhat Eisenstein series minus its constant term
AutomorphicForm.bruhatEisenstein_sub_constantTerm_isRapidlyDecreasingOn20 below · cited by 1 · depth 30 - Split fibre integral preserves smoothness and compact support
AutomorphicForm.contDiff_splitFibreIntegral_psiGL_complex0 below · cited by 1 · depth 30 - Smoothness and compact support of the split fibre integral over ℝ
AutomorphicForm.contDiff_splitFibreIntegral_psiGL_real0 below · cited by 1 · depth 30 - Continuity of right convolution of an automorphic L² function
AutomorphicForm.continuous_convOp_of_isAutomorphicFnAt_canonicalTruncationDomain_of_continuous26 below · cited by 7 · depth 30 - Finite local product: continuity, fractional-ideal support, polynomial growth
AutomorphicForm.continuous_finprod_localFactor_and_exists_fractionalIdeal_norm_finprod_le_of_isCompact1 below · cited by 1 · depth 30 - Idelic base change: continuity, norm, principal ideles, σ-fixed ideles
AutomorphicForm.continuous_injective_norm_pow_principal_range_eq_fixed_unitsMap_genuineBaseChange5 below · cited by 2 · depth 30 - Continuity of the quasi-characters inducing a nonzero section
AutomorphicForm.continuous_of_isInducedSection_of_continuous_of_apply_ne_zero0 below · cited by 1 · depth 30 - Right convolution splits along an a.e. decomposition of automorphic functions
AutomorphicForm.convOp_eq_add_add_of_ae_eq_restrict_canonicalTruncationDomain_of_isAutomorphicFnAt_of_continuous27 below · cited by 5 · depth 30 - Right convolution stabilises cut isotypic cusp spaces (slab domain)
AutomorphicForm.convOp_mem_isotypicCuspSubmodule_inf_archCutSubmodule_levelOne_of_conjInvariant_of_isFundamentalDomain_slab23 below · cited by 1 · depth 30 - Equally normalised Haar measures are coupled at y=1
AutomorphicForm.coupled_one_of_forall_integral_centralizer_eq_mul_of_forall_integral_twistedCentralizer_eq_mul0 below · cited by 1 · depth 30 - Local matching at a split regular norm pair of prime degree
AutomorphicForm.eq_of_isTwistedOrbitalIntegral_of_isOrbitalIntegral_diagUnits2_of_areMatchingLocal_of_measure_eq_one_of_prime3 below · cited by 2 · depth 30 - Uniform coordinate bound for flat induced-section families
AutomorphicForm.exists_basis_forall_flat_isInducedSection_family_eq_sum_and_norm_sq_le_lintegral_of_principalLevel_archCutSubmodule32 below · cited by 2 · depth 30 - Uniform transfer constant for twisted hyperbolic intercepts
AutomorphicForm.exists_const_forall_exists_windingDatum_hyperbolicIntercept_sub_finrank_mul_const_mul_sum_eq_sum_satakeLaurent_mul_coeff_of_eq_affine_of_areMatchingArch_of_areMatchingLocal1,445 below · cited by 1 · depth 30 - Existence of semi-locally factorised test functions on GL₂(A_L)
AutomorphicForm.exists_continuous_hasCompactSupport_isSemiLocalFactorization_and_union_of_isArchTestFactor_of_isSemiLocalTestFn1 below · cited by 3 · depth 30 - Continuous section functions and single-valued orbital integrals at regular semisimple γ
AutomorphicForm.exists_continuous_isSectionFnOn_and_isOrbitalIntegralOn_iff_completion_of_isRegularSemisimple2 below · cited by 2 · depth 30 - Continuous section functions and the orbital integral formula over a field
AutomorphicForm.exists_continuous_isSectionFnOn_and_isOrbitalIntegralOn_iff_of_isRegularSemisimple_of_field2 below · cited by 10 · depth 30 - Matched split pair with equal non-zero twisted orbital integral
AutomorphicForm.exists_diagUnits2_normString_isOrbitalIntegral_ne_zero_isTwistedOrbitalIntegral_eq_heckeWord_of_ramificationIdx_eq_one_of_prime102 below · cited by 1 · depth 30 - Diagonal representatives over K and the norm map on σ-classes
AutomorphicForm.exists_diagonal_classReps_and_normMap_injOn_of_pairwise_disjoint_sigmaClasses4 below · cited by 2 · depth 30 - Norm string of a diagonal datum over cyclic L/K
AutomorphicForm.exists_diagonal_normString_eq_toTensorGL_globalPoints_of_baseChangeGL_eq_globalPoints0 below · cited by 4 · depth 30 - Coupled elliptic family for a negative central twisted class
AutomorphicForm.exists_elliptic_family_coupled_inf_twistedCentralizer_conjAe_of_neg1 below · cited by 2 · depth 30 - Entire Whittaker coefficients of the GL₂ Bruhat Eisenstein family
AutomorphicForm.exists_entire_whittakerCoefficient_bruhatEisenstein_eq_eulerProduct_mul_summable_norm_tsum_le_rpow_neg_of_isArchKFinite_family_of_unitary89 below · cited by 1 · depth 30 - Arch-K-finite forms on K_∞ as sums of local products
AutomorphicForm.exists_eq_sum_prod_archComponent_of_isArchKFinite0 below · cited by 1 · depth 30 - Arthur's parametrix lemma for factorizable test functions on GL₂
AutomorphicForm.exists_eq_sum_rightConv_conjInvariant_principalLevel_of_isFactorizableTestFn9 below · cited by 1 · depth 30 - Unramified uniformiser stays a uniformiser of the semi-local algebra
AutomorphicForm.exists_eq_tmul_mul_of_mul_self_eq_and_exists_isIdempotentElem_of_not_mem_semiLocalIntegers0 below · cited by 1 · depth 30 - Hyperbolic slope and intercept as sums of orbital integrals
AutomorphicForm.exists_finset_forall_slope_eq_sum_classIntegral_and_intercept_eq_sum_weightedClassIntegral_of_hyperbolicTerm_eq_affine225 below · cited by 3 · depth 30 - Twisted hyperbolic slope and intercept as twisted orbital class sums
AutomorphicForm.exists_finset_forall_slope_eq_sum_twistedClassIntegral_and_intercept_eq_sum_weightedTwistedClassIntegral_haarQuotient_of_eq_affine241 below · cited by 2 · depth 30 - SU(2)-type control at a complex place in L²
AutomorphicForm.exists_forall_eLpNorm_archDerivAtComplex_iH_and_E_sub_Fm_and_iE_add_iFm_foldr_le_of_mem_archCutSubmodule40 below · cited by 1 · depth 30 - Weight bound for E-F on archimedean derivative words
AutomorphicForm.exists_forall_eLpNorm_archDerivAt_E_sub_Fm_foldr_le_of_mem_archCutSubmodule36 below · cited by 1 · depth 30 - Twisted Maass–Selberg relations for truncated Eisenstein series
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_axis_continuation_mul_conj_lambdaT_sigmaAdelicAct_eq_maassSelberg_cases_slab_of_flat288 below · cited by 1 · depth 30 - Truncated defect of a hyperbolic twisted class integrates to zero
AutomorphicForm.exists_forall_integrableOn_indicator_mul_setIntegral_finsum_borel_sigmaConjClassOrbit_sub_setIntegral_constantTerm_and_setIntegral_eq_zero104 below · cited by 1 · depth 30 - Truncated Eisenstein series as pseudo-Eisenstein series of the truncated section
AutomorphicForm.exists_forall_lambdaT_pseudoEisenstein_eq_pseudoEisenstein_ite_adelicHeight_le_of_mem_canonicalTruncationDomain21 below · cited by 3 · depth 30 - Archimedean twisted descent for ℂ/ℝ at a real scalar
AutomorphicForm.exists_forall_nhds_one_isOrbitalIntegralOn_of_isTwistedOrbitalIntegralOn_conjAe_toTensorGL_mul_scalar15 below · cited by 1 · depth 30 - Uniform axis bound for a normalised archimedean Γ-factor
AutomorphicForm.exists_forall_norm_archIntertwiningComplex_le_and_norm_deriv_le_of_re_eq_zero0 below · cited by 1 · depth 30 - Uniform moderate growth of GL₂ Eisenstein series on centre-cut Siegel sets
AutomorphicForm.exists_forall_norm_axis_continuation_le_mul_pow_archParam_weight_mul_adelicHeight_rpow_of_mem_centreCutSiegelSet_mul_of_flat410 below · cited by 1 · depth 30 - Uniform rapid decay of non-constant part of GL₂ Eisenstein series
AutomorphicForm.exists_forall_norm_axis_continuation_sub_constantTerm_le_mul_pow_archParam_weight_mul_rpow_neg_of_isCompact_of_flat261 below · cited by 4 · depth 30 - Polynomial bound for constant terms of flat unitary Eisenstein families
AutomorphicForm.exists_forall_norm_constantTerm_axis_continuation_le_mul_pow_archParam_weight_mul_adelicHeight_rpow_half_of_flat286 below · cited by 3 · depth 30 - Right convolution bounds L² of a slab domain by sup on compacta
AutomorphicForm.exists_forall_norm_rightConv_le_mul_eLpNorm_of_isLsXiFunction_of_isCompact_of_isFundamentalDomain_slab16 below · cited by 1 · depth 30 - Continuous block of the GL₂ spectral expansion on A× B
AutomorphicForm.exists_forall_setIntegral_convOp_continuousProjection_eq_mul_setIntegral_prod_tsum_integral_sum_rightConv_axis_continuation1,066 below · cited by 1 · depth 30 - Integrated truncated twisted kernel and its continuous-spectrum expansion
AutomorphicForm.exists_forall_setIntegral_lambdaT_finsum_sub_lambdaT_tsum_sub_lambdaT_finsum_chiDet_sigmaAdelicAct_symm_eq_mul_tsum_integral_sum_rightConv_mul_setIntegral_lambdaT_mul_conj_lambdaT_of_norm_eq_one1,249 below · cited by 1 · depth 30 - Unfolding Borel-coset sums into Iwasawa coordinates
AutomorphicForm.exists_forall_setLIntegral_tsum_borelSubgroup_cosets_eq_mul_lintegral_iwasawa15 below · cited by 4 · depth 30 - Uniform bound for twisted orbital integrals of one translate
AutomorphicForm.exists_forall_sum_lintegral_orbital_add_weightedOrbital_le_of_isSemiLocalFactorization_indicator_translate280 below · cited by 1 · depth 30 - Twisted-class truncation weights collapse onto the Borel part
AutomorphicForm.exists_forall_tsum_indicator_add_indicator_weyl_mul_integral_eq_indicator_mul_setIntegral_mul_finsum_borel_sigmaConjClassOrbit8 below · cited by 1 · depth 30 - Shalika germ expansion at a scalar on GL₂(Kᵥ)
AutomorphicForm.exists_germ_forall_isOrbitalIntegral_eq_add_nhds_scalar_of_isLocalTestFn20 below · cited by 1 · depth 30 - Adelic GL₂ points with prescribed components
AutomorphicForm.exists_glArch_eq_and_finComponent_glFin_eq_and_mem_localIntegralSet0 below · cited by 1 · depth 30 - Prescribing archimedean and semi-local components of adelic GL₂
AutomorphicForm.exists_glArch_eq_and_semiLocalComponent_glFin_eq_and_mem_semiLocalIntegralSet0 below · cited by 1 · depth 30 - Transport of the Gram normalisation along a ring isomorphism
AutomorphicForm.exists_gram_map_of_ringEquiv_of_exists_gram0 below · cited by 1 · depth 30 - Properness of regular hyperbolic twisted GL₂-orbit maps adelically
AutomorphicForm.exists_isCompact_forall_exists_mem_mul_of_inv_mul_globalPoints_mul_sigmaAdelicAct_centralScalar_mul_mem_of_diagonal45 below · cited by 6 · depth 30 - Local Haar domination in archimedean flow-chart coordinates
AutomorphicForm.exists_isHaarMeasure_lintegral_comp_glArch_flowChart_mul_le3 below · cited by 1 · depth 30 - Gram normalisation couples central measures at a split complex place
AutomorphicForm.exists_isNormConjugator_and_coupled_of_gram_of_algHom_complex0 below · cited by 1 · depth 30 - Gram normalisation couples centralizer measures at a split real place
AutomorphicForm.exists_isNormConjugator_and_coupled_of_gram_of_algHom_real0 below · cited by 1 · depth 30 - Scalar σ-conjugacy at a real place holds iff c>0
AutomorphicForm.exists_isSigmaConjugate_scalar_conjAe_iff_pos_of_isNormConjugator_scalar0 below · cited by 2 · depth 30 - Coupled measures force σ-conjugacy of δ to a scalar
AutomorphicForm.exists_isSigmaConjugate_scalar_of_coupled0 below · cited by 3 · depth 30 - Norm conjugate to a scalar implies σ-conjugate to a scalar
AutomorphicForm.exists_isSigmaConjugate_scalar_of_isNormConjugator_scalar_of_algHom_of_prime0 below · cited by 2 · depth 30 - Existence of twisted orbital integrals over ℂ
AutomorphicForm.exists_isTwistedOrbitalIntegralOn_of_isNormConjugator_of_isRegularSemisimple_complex4 below · cited by 1 · depth 30 - Existence of twisted orbital integrals at a real place
AutomorphicForm.exists_isTwistedOrbitalIntegralOn_of_isNormConjugator_of_isRegularSemisimple_real4 below · cited by 1 · depth 30 - Continuous twisted sections at an archimedean place
AutomorphicForm.exists_isTwistedSectionFnOn_and_continuous_completion_of_isRegularSemisimple_normString3 below · cited by 2 · depth 30 - Continuous twisted sections at a central class, archimedean place
AutomorphicForm.exists_isTwistedSectionFnOn_and_continuous_completion_of_isSigmaConjugate_scalar_of_prime13 below · cited by 2 · depth 30 - Continuous twisted section at a non-σ-conjugate scalar datum
AutomorphicForm.exists_isTwistedSectionFnOn_and_continuous_completion_of_not_isSigmaConjugate_scalar_of_finrank_eq_two9 below · cited by 1 · depth 30 - Continuous twisted section functions for complex conjugation on GL₂
AutomorphicForm.exists_isTwistedSectionFnOn_and_continuous_conjAe_of_isRegularSemisimple_normString0 below · cited by 2 · depth 30 - Test function with prescribed residue of a twisted orbital zeta integral
AutomorphicForm.exists_mem_schwartzBruhat2_tendsto_sub_one_mul_lintegral_twistedCentralizer_nhdsGT_one_of_forall_integral_eq_mul_prod_integral_of_forall_ne_scalar_of_finrank_eq_two412 below · cited by 1 · depth 30 - GL₂(L⊗_K K_∞) splits over the infinite places
AutomorphicForm.exists_mulEquiv_generalLinearGroup_tensorProduct_infiniteAdeleRing_pi0 below · cited by 4 · depth 30 - Twisted centralisers near real scalars and coupled Haar measures
AutomorphicForm.exists_nhds_one_forall_exists_isHaarMeasure_coupled_toTensorGL_mul_scalar_one2 below · cited by 1 · depth 30 - Twisted centraliser of a regular diagonal base-change element: Haar measure comparison
AutomorphicForm.exists_pos_forall_exists_isHaarMeasure_twistedCentralizer_integral_eq_mul_integral_prod_toTensorGL_diagUnits21 below · cited by 3 · depth 30 - Explicit constant in the finite-place twisted orbital comparison
AutomorphicForm.exists_pos_forall_integral_eq_and_forall_nhds_isTwistedOrbitalIntegral_mul_eq_mul_of_not_isSigmaConjugate_scalar_of_finrank_eq_two6 below · cited by 1 · depth 30 - Split-place transfer of twisted orbital integrals for GL₂
AutomorphicForm.exists_pos_forall_isOrbitalIntegralOn_smul_splitFibreIntegral_of_isTwistedOrbitalIntegralOn_of_algHom_of_prime_of_forall_exists_isSectionFnOn_of_isMulRightInvariant5 below · cited by 2 · depth 30 - Twisted orbital integrals tend to κ I' for c<0
AutomorphicForm.exists_pos_forall_tendsto_isTwistedOrbitalIntegralOn_mul_nhdsGT_conjAe_of_neg_of_inf_twistedCentralizer11 below · cited by 2 · depth 30 - Archimedean base change splits over the infinite places
AutomorphicForm.exists_ringEquiv_tensorProduct_infiniteAdeleRing_pi_forall_tmul0 below · cited by 4 · depth 30 - Base change of L⊗_K A along a bicontinuous ring isomorphism
AutomorphicForm.exists_ringEquiv_tensor_baseChange_of_ringEquiv0 below · cited by 2 · depth 30 - Ramified real place model: L⊗_K Kᵥ≅ℂ⊗_ℝℝ
AutomorphicForm.exists_ringEquiv_tensor_completion_complex_of_isRamified0 below · cited by 4 · depth 30 - Square-root cutoff for conjugation classes on GL₂(ℝ)
AutomorphicForm.exists_smooth_hasCompactSupport_forall_nhds_one_apply_conj_eq_apply_conj_mul_scalar_sq0 below · cited by 1 · depth 30 - Matching descends to any factorisations, up to reciprocal scalars
AutomorphicForm.exists_smul_areMatchingArch_and_areMatchingLocal_of_areMatchingAt_of_isSemiLocalFactorization_of_isUnitFactorization2 below · cited by 1 · depth 30 - Semi-local factors of a non-zero test function are unique up to scalars
AutomorphicForm.exists_smul_eq_of_isSemiLocalFactorization_of_isSemiLocalFactorization_of_exists_ne_zero0 below · cited by 2 · depth 30 - Uniqueness of unit factorisations up to scalars of product one
AutomorphicForm.exists_smul_eq_of_isUnitFactorization_of_isUnitFactorization_of_exists_ne_zero0 below · cited by 2 · depth 30 - Factorizable test functions as sums of unit-factorized ones
AutomorphicForm.exists_sum_isUnitFactorization_of_isFactorizableTestFn4 below · cited by 1 · depth 30 - Archimedean K-finite induced vectors as local zeta sections
AutomorphicForm.exists_sum_mul_localZeta_bottomRow_eq_of_rightTranslatesSpanFinite9 below · cited by 1 · depth 30 - Summable dominants and Lipschitz bounds for twisted Maass–Selberg pairings
AutomorphicForm.exists_summable_dominant_rightConv_axis_family_sigma_maassSelberg_pairings_of_isSemiLocalFactorization_lipschitz426 below · cited by 1 · depth 30 - Uniform Haar families on split-class centralisers, one factorisation constant
AutomorphicForm.exists_torusFamily_centralScalar_mul_diagUnits2_coupled_massOne_restrictedProduct5 below · cited by 1 · depth 30 - The K-side class sum as a winding-datum coefficient array
AutomorphicForm.exists_windingDatum_forall_coeff_eq_mul_finsum_mul_prod_zpow_neg_mul_ideleNorm_mul_integral_orbital_of_smul_eq_map_partAt_of_ne_one_unweighted74 below · cited by 1 · depth 30 - Moving one chart coordinate: right translation by a conjugated flow
AutomorphicForm.flowChart_add_single_eq_mul_conj0 below · cited by 1 · depth 30 - Torus constant c_H: lower-integral form implies Bochner form
AutomorphicForm.forall_integral_sigmaCentraliser_eq_mul_integral_prod_centralScalar_mul_baseChangeGL_diagUnits2_of_forall_lintegral_eq_idelesBaseChange25 below · cited by 2 · depth 30 - Archimedean component of scalar(z)cdotdiag(a,b)
AutomorphicForm.glArch_centralScalar_mul_diagUnits20 below · cited by 15 · depth 30 - Derivative along a conjugated one-parameter flow at a complex place
AutomorphicForm.hasDerivAt_apply_mul_archComplexGLAt_inv_mul_archFlowMatrixComplex_mul1 below · cited by 3 · depth 30 - Directional derivative along a conjugated one-parameter subgroup
AutomorphicForm.hasDerivAt_apply_mul_archRealGLAt_inv_mul_archFlowMatrix_mul1 below · cited by 3 · depth 30 - Archimedean sign: Harish-Chandra and Weil constants multiply to -1
AutomorphicForm.hcConst_mul_weilConst_mul_eq_neg_one_of_gram_conjAe_of_coupled_of_neg12 below · cited by 2 · depth 30 - Twisted and untwisted truncated cuspidal kernels integrate equally
AutomorphicForm.integrableOn_and_setIntegral_lambdaT_tsum_finsum_twistedConvOp_mul_conj_eq_setIntegral_lambdaT_tsum_convOp_mul_conj_sigmaAdelicAct_symm542 below · cited by 1 · depth 30 - Constant term of an upper-triangular twisted class as orbital integrals
AutomorphicForm.integrableOn_and_setIntegral_mul_constantTerm_finsum_borelSigmaConjClassOrbit_eq_inv_measure_mul_tsum_integral_integral8 below · cited by 1 · depth 30 - Unfolding a class sum on adelic GL₂ over Hbackslash G
AutomorphicForm.integrableOn_tsum_and_setIntegral_tsum_comp_globalPoints_inv_mul_eq_integral_haarQuotient_setIntegral8 below · cited by 2 · depth 30 - Modulus-one change of variables for twisted unipotent orbital integrals
AutomorphicForm.integral_integral_unipotentGL2_conj_twistedOrbital_eq_integral_mul_integral_and_lintegral_lt_top_of_norm_div_ne_one4 below · cited by 1 · depth 30 - Adjointness of the Weyl intertwining integral under a Galois twist
AutomorphicForm.integral_mul_conj_weylIntertwiningIntegral_sigmaAdelicAct_eq_of_sigmaInvariant_and_of_sigmaReversed_of_principalLevel_of_ne_bot157 below · cited by 1 · depth 30 - Windowed Iwasawa factorisation for induced sections on GL₂
AutomorphicForm.integral_rationalTorusUnipotentQuotient_section_mul_conj_eq_mul_setIntegral_iwasawa_of_window21 below · cited by 3 · depth 30 - Flow derivatives at a complex place: smoothness, linearity, brackets
AutomorphicForm.isArchSmoothAtComplex_archDerivAtComplex_and_add_and_smul_and_comm0 below · cited by 8 · depth 30 - Casimirs of a right convolution at a complex place, principal level
AutomorphicForm.isArchSmoothAtComplex_rightConv_and_exists_archCasimirAtComplex_rightConv_eq_of_isArchBiFinite_principal15 below · cited by 1 · depth 30 - Casimir of a right convolution at principal level K(N)
AutomorphicForm.isArchSmoothAt_rightConv_and_exists_archCasimirAt_rightConv_eq_of_isArchBiFinite_principal14 below · cited by 1 · depth 30 - Automorphisation of a bounded compactly supported test function
AutomorphicForm.isAutomorphicFnAt_finsum_integral_indicator_canonicalTruncationDomain21 below · cited by 4 · depth 30 - Galois transport of level, conjugation invariance and support of adelic kernels
AutomorphicForm.isBiInvariantUnder_principalLevel_comap_and_conjInvariant_comp_sigmaAdelicAct6 below · cited by 1 · depth 30 - Transport of torus-shell integrals to Ω_L×Ω_K
AutomorphicForm.isFundamentalDomain_image_and_forall_setLIntegral_torusShell_eq_mul_setLIntegral_prod26 below · cited by 1 · depth 30 - Haar measures transported along a bicontinuous ring isomorphism
AutomorphicForm.isHaarMeasure_map_and_exists_isHaarMeasure_centralizer_of_ringEquiv0 below · cited by 2 · depth 30 - Right convolution preserves Hecke eigenvalues away from the levels
AutomorphicForm.isHeckeCosetEigenfunctionAt_rightConv_of_isBiInvariantUnder_levelOne_of_not_dvd8 below · cited by 1 · depth 30 - Inversion-invariance transfers along coupled centraliser measures
AutomorphicForm.isInvInvariant_of_coupled_of_isInvInvariant0 below · cited by 5 · depth 30 - Inverse-invariance of Haar measure on archimedean twisted centralizers, scalar norm
AutomorphicForm.isInvInvariant_twistedCentralizer_infiniteAdeleRing_of_isNormConjugator_scalar7 below · cited by 1 · depth 30 - Hecke-word indicator sums are local test functions
AutomorphicForm.isLocalTestFn_sum_indicator_localIntegralSet_prod_mul_zpow_inv_mul0 below · cited by 9 · depth 30 - Transport of norm-conjugators and coupled measures along ring isomorphisms
AutomorphicForm.isNormConjugator_map_iff_and_coupled_map_iff_of_ringEquiv0 below · cited by 2 · depth 30 - Transport of the GL₂ orbital-integral relation along a ring isomorphism
AutomorphicForm.isOrbitalIntegralOn_map_generalLinearGroup_map_of_ringEquiv0 below · cited by 2 · depth 30 - Twisted orbital integral at a negative central class: sign -1
AutomorphicForm.isOrbitalIntegralOn_scalar_neg_of_isTwistedOrbitalIntegralOn_conjAe_of_gram_of_nhds_forall_isNormConjugator_of_neg41 below · cited by 1 · depth 30 - Linearity and measurability of adelic orbital integrals
AutomorphicForm.isOrbitalIntegralOn_sum_mul_centralScalar_mul_and_measurable_of_isRegularSemisimple7 below · cited by 1 · depth 30 - Regular semisimplicity of local components of z diag(u,1)
AutomorphicForm.isRegularSemisimple_finComponent_glFin_centralScalar_mul_diagUnits2_of_ne_one0 below · cited by 8 · depth 30 - Regular semisimplicity of the norm string of a diagonal global class
AutomorphicForm.isRegularSemisimple_normString_of_baseChangeGL_eq_globalPoints_of_norm_ne_one0 below · cited by 4 · depth 30 - Right convolution preserves cusp forms and yields smooth vectors
AutomorphicForm.isSmoothCuspAutomorphicFnAt_rightConv_of_isFundamentalDomain_slab11 below · cited by 1 · depth 30 - Transport of twisted orbital integrals along a ring isomorphism
AutomorphicForm.isTwistedOrbitalIntegralOn_map_of_ringEquiv_and_isHaarMeasure_map0 below · cited by 2 · depth 30 - Quotient of two idele class characters with equal weights
AutomorphicForm.isUnramifiedCharAt_and_archLocalChar_mul_inv_eq_cpow_of_archLocalChar_eq_of_localChar_eq4 below · cited by 1 · depth 30 - Twisted truncated kernel versus untwisted ξ₀-kernel
AutomorphicForm.lambdaT_finsum_integral_sigmaAdelicAct_eq_and_lambdaT_finsum_twistedConvOp_chiDet_eq_and_rightConv_mul_ideleNorm_det_rpow_eq28 below · cited by 1 · depth 30 - Convergence of twisted orbital integrals at a regular diagonal class
AutomorphicForm.lintegral_abs_twistedOrbital_lt_top_and_integrable_norm_and_weighted_and_exists_height_mul_le_of_zpowers75 below · cited by 1 · depth 30 - Finiteness of the truncated Rankin–Selberg integrand on the slab
AutomorphicForm.lintegral_canonicalTruncationDomain_enorm_pseudoEisenstein_mul_enorm_truncatedSection_add_tsum_lt_top_of_re_lt_re55 below · cited by 3 · depth 30 - Subadditivity of twisted orbital and height-weighted orbital integrals
AutomorphicForm.lintegral_orbital_le_sum_and_weightedOrbital_le_sum_of_isSemiLocalFactorization_of_eq_sum1 below · cited by 1 · depth 30 - Three-way decomposition of automorphic L² in a determinant slab
AutomorphicForm.lsXi_threeWay_orthogonal_decomposition_haar_ae_of_isFundamentalDomain_slab12 below · cited by 3 · depth 30 - Transport of the archimedean Gram normalisation along Xi
AutomorphicForm.map_centralizer_map_ringEquiv_pi_eq_smul_gram_of_infiniteAdeleRing0 below · cited by 1 · depth 30 - Transport of the archimedean Gram normalisation along Xi
AutomorphicForm.map_twistedCentralizer_map_ringEquiv_pi_eq_smul_gram_of_infiniteAdeleRing0 below · cited by 1 · depth 30 - Unipotent integration formula for GL₂(A_K)
AutomorphicForm.measure_pow_three_mul_measure_mul_lintegral_mul_apply_col_det_eq_mul_dedekindZeta_two_mul_lintegral_of_forall_lintegral_mul_unipotentGL2_eq_one29 below · cited by 1 · depth 30 - Central translates that are twisted norms yield idelic norms
AutomorphicForm.mem_range_idelicNorm_of_isNormOf_centralScalar_mul_globalPoints_diagUnits2_of_mem_range_norm3 below · cited by 1 · depth 30 - Twisted orbit parametrised by coset representatives and central scalars
AutomorphicForm.mem_sigmaConjClassOrbit_and_existsUnique_and_transport_of_leftCosetRepresentatives0 below · cited by 2 · depth 30 - Coordinatewise span criterion after the archimedean splitting Xi
AutomorphicForm.mem_span_map_ringEquiv_pi_iff_of_span_eq_range_includeRight0 below · cited by 1 · depth 30 - Coordinatewise splitting of the archimedean twisted commutant
AutomorphicForm.mem_span_map_ringEquiv_pi_iff_of_span_eq_setOf_mul_eq_mul_map_sigmaTensor0 below · cited by 1 · depth 30 - Twisted stabiliser of a regular diagonal element of GL₂
AutomorphicForm.mem_twistedStabilizer_iff_diagonal_or_antidiagonal2 below · cited by 1 · depth 30 - Evaluation of the elliptic germ constant A ν_T(T_c)
AutomorphicForm.mul_measureReal_torusUnits_eq_neg_div_of_forall_isOrbitalIntegral_eq_add_nhds_scalar_of_forall_not_diagonal24 below · cited by 1 · depth 30 - Diagonal classes with non-norm ratio admit no twisted norm
AutomorphicForm.not_exists_isNormOf_centralScalar_mul_globalPoints_of_div_not_mem_range_norm127 below · cited by 1 · depth 30 - Coupled pair at a scalar norm: c>0 and transfer to √c 1
AutomorphicForm.pos_and_exists_coupled_toTensorGL_scalar_one_iff_of_coupled_scalar_conjAe0 below · cited by 3 · depth 30 - Weil: GL₂ slab rate versus idelic rate and covolume
AutomorphicForm.rate_eq_mul_rate_mul_measure_pow_three_of_forall_lintegral_mul_apply_col_det_eq_mul_lintegral_of_forall_isFundamentalDomain_op_inter_ideleNorm_det_Icc50 below · cited by 1 · depth 30 - Hecke-word eigenvalue for adelic induced sections under right convolution
AutomorphicForm.rightConv_eq_prod_pow_mul_pow_mul_rightConv_of_isInducedSection_of_isSemiLocalFactorization4 below · cited by 1 · depth 30 - Semi-local central transfer from one-place central comparisons
AutomorphicForm.semilocal_central_transfer_of_forall_oneplace_of_isInvInvariant7 below · cited by 2 · depth 30 - Semi-local central transfer with reference measures and per-place factors
AutomorphicForm.semilocal_central_transfer_of_forall_oneplace_of_referenceMeasures7 below · cited by 1 · depth 30 - Cuspidal block of the rectangle spectral expansion for GL₂
AutomorphicForm.setIntegral_convOp_cuspProjection_eq_mul_setIntegral_prod_tsum_convOp_mul_conj_of_orthonormal_isotypicCuspSubmodule535 below · cited by 1 · depth 30 - Residual block of the rectangular GL₂ spectral expansion
AutomorphicForm.setIntegral_convOp_residualProjection_eq_mul_setIntegral_prod_finsum_chiDet_mul_chiDet_inv86 below · cited by 1 · depth 30 - Unfolding the centre-folded GL₂ kernel against a truncated test function
AutomorphicForm.setIntegral_finsum_integral_centralScalar_mul_eq_convOp_finsum_integral_indicator_of_hasCompactSupport8 below · cited by 1 · depth 30 - Unfolding a pseudo-Eisenstein series against an automorphic function
AutomorphicForm.setIntegral_mul_pseudoEisenstein_eq_integral_rationalTorusUnipotentQuotient_constantTerm_mul20 below · cited by 4 · depth 30 - Adjoint of right convolution on a slab fundamental domain
AutomorphicForm.setIntegral_rightConv_mul_conj_eq_setIntegral_mul_conj_rightConv_flat_of_isLsXiFunction_of_isFundamentalDomain_slab15 below · cited by 2 · depth 30 - Exact value of the torus-shell integral over Ω_L×Ω_K
AutomorphicForm.setLIntegral_prod_torusShell_eq_and_setIntegral_prod_torusShell_eq26 below · cited by 1 · depth 30 - Unfolding a coset sum on Φ₀ to HbackslashGL₂(A_L)
AutomorphicForm.setLIntegral_tsum_comp_globalPoints_inv_mul_eq_lintegral_haarQuotient_setLIntegral_of_subgroup6 below · cited by 1 · depth 30 - Slope comparison of hyperbolic class sums under base change
AutomorphicForm.sum_mul_integral_haarQuotient_ker_idelicNorm_eq_slopeFactor_mul_sum_sum_mul_integral_of_forall_eq_mul_comp_idelicNorm1 below · cited by 2 · depth 30 - Hyperbolic class sums of a Hecke word as winding pairing
AutomorphicForm.sum_slotFamilyCoeff_mul_sum_mul_integral_orbital_eq_sum_prod_mul_windingDatum_coeff_of_forall_coeff_eq_of_smul_eq_map_partAt_of_ne_one_unweighted76 below · cited by 1 · depth 30 - Residue at s=1 of a quaternionic adelic zeta integral
AutomorphicForm.tendsto_sub_one_mul_lintegral_mul_ideleNorm_det_rpow_twistedCentralizer_nhdsGT_one_of_isFundamentalDomain_of_forall_ne_scalar_of_finrank_eq_two49 below · cited by 1 · depth 30 - Central twisted orbital integral equals minus the scalar orbital integral
AutomorphicForm.twistedOrbitalIntegral_eq_neg_orbitalIntegral_scalar_of_forall_germ_of_forall_germValue_of_forall_nhds_mul_of_not_isSigmaConjugate_of_finrank_eq_two42 below · cited by 1 · depth 30 - Meromorphic continuation of the Weyl intertwining integral
AutomorphicForm.weylIntertwiningIntegral_meromorphicOn_of_isInducedSection_family78 below · cited by 7 · depth 30 - Automorphic functions are a.e. strongly measurable for adelic Haar measure
AutomorphicForm.aestronglyMeasurable_adelicGLHaar_of_isAutomorphicFnAt_slab12 below · cited by 21 · depth 31 - Vanishing limit formula on GL₂(ℂ) at a scalar
AutomorphicForm.apply_scalar_eq_zero_of_conj_unitary_eq_of_forall_integral_upperTriangular_complex_eq_zero0 below · cited by 1 · depth 31 - Circle weight n at a complex place forces iH-eigenvalue in
AutomorphicForm.archDerivAtComplex_iH_eq_smul_of_hasCircleWeightAt_of_isArchSmoothAtComplex0 below · cited by 1 · depth 31 - Rational Borel invariance of the continued adelic Eisenstein series
AutomorphicForm.axis_continuation_globalPoints_mul_eq_of_mem_borelSubgroup_of_isIdeleClassChar4 below · cited by 1 · depth 31 - Semi-local coordinates: local embedding, integrality, maximal compact
AutomorphicForm.baseChangeAlgEquiv_semiLocalComponent_localEmbed_and_mem_semiLocalIntegers_iff_and_semiLocalComponent_mem_of_mem_adelicMaximalCompact0 below · cited by 3 · depth 31 - Base change along K/K is the identity on GL₂(A_K)
AutomorphicForm.baseChangeGL_toTensorGL_self0 below · cited by 1 · depth 31 - Parametrisation of a regular twisted conjugacy class in GL₂
AutomorphicForm.bijOn_mul_map_unipotentGL2_mul_scalar_borelSigmaConjClass_of_norm_div_ne_one0 below · cited by 1 · depth 31 - Closed form of the central–elliptic base-change comparison constant
AutomorphicForm.centralEllipticConstant_eq_of_factorization_of_normFibre_of_exists_ne_zero909 below · cited by 1 · depth 31 - Centralisers in adelic GL₂ as a restricted product
AutomorphicForm.centralizer_secondCountableTopology_locallyCompactSpace_and_exists_glArch_finComponent_localIntegralSet_isOpen_surjective_isCompact_restrictedProduct1 below · cited by 7 · depth 31 - v-component of a central local unit times diag(u,1)
AutomorphicForm.coe_finComponent_glFin_centralScalar_localUnit_mul_diagUnits20 below · cited by 2 · depth 31 - Constant term and truncation commute with the Galois twist
AutomorphicForm.constantTerm_sigmaSectionActOn_and_lambdaT_sigmaSectionActOn7 below · cited by 3 · depth 31 - Smoothness and compact support of an archimedean twisted integral
AutomorphicForm.contDiff_and_hasCompactSupport_integral_mul_comp_conjAe_toTensorGL_mul_scalar1 below · cited by 1 · depth 31 - Unit-normalised torus measures are coupled along y=1
AutomorphicForm.coupled_one_diagUnits2_of_normString_eq_toTensorGL_of_measure_eq_one2 below · cited by 2 · depth 31 - Determinant band has equal mass on coupled tori
AutomorphicForm.detBand_eq_inf_twistedCentralizer_detBand_and_pos_and_lt_top_of_coupled0 below · cited by 1 · depth 31 - L² bound for weighted sums of pure-weight pieces
AutomorphicForm.eLpNorm_sum_weight_smul_le_mul_eLpNorm_sum_of_hasArchCharacterAtZero22 below · cited by 1 · depth 31 - Weighted L² bound for circle-weight packets at a complex place
AutomorphicForm.eLpNorm_sum_weight_smul_le_mul_eLpNorm_sum_of_hasCircleWeightAt23 below · cited by 1 · depth 31 - Induced sections agree when they agree on the maximal compact
AutomorphicForm.eq_of_isInducedSection_of_forall_adelicMaximalCompact_eq2 below · cited by 6 · depth 31 - Independence of the local orbital integral from the section function
AutomorphicForm.eq_of_isOrbitalIntegral_of_isOrbitalIntegral_of_isRegularSemisimple3 below · cited by 5 · depth 31 - Finitely spanned K-finite induced sections at trivial level vanish
AutomorphicForm.eq_zero_of_isInducedSection_of_isArchKFinite_of_forall_mem_span_range_of_principalLevel_bot3 below · cited by 1 · depth 31 - Unique unipotent σ-twisted diagonalisation when N(a/b)≠ 1
AutomorphicForm.existsUnique_sigmaConj_unipotentGL2_apply_zero_one_eq_zero_of_norm_div_ne_one0 below · cited by 1 · depth 31 - Admissible flat family with prescribed maximal-compact values
AutomorphicForm.exists_admissible_flat_family_restrict_eq_of_sameClass_of_principalLevel_archCutSubmodule11 below · cited by 2 · depth 31 - Compact directions F-E and i(E+F) are iH-conjugates
AutomorphicForm.exists_archDerivAtComplex_Fm_sub_E_and_iE_add_iFm_eq_rightTranslate_iH_rightTranslate0 below · cited by 1 · depth 31 - Conjugation-invariant parametrix kernels on the archimedean matrix algebra
AutomorphicForm.exists_conjInvariant_forall_exists_eq_sum_integral_comp_mul_archEntries7 below · cited by 1 · depth 31 - Orbital integrals of a smooth family depend smoothly on the parameter
AutomorphicForm.exists_contDiff_hasCompactSupport_forall_isOrbitalIntegralOn_slice_of_isRegularSemisimple2 below · cited by 2 · depth 31 - Elliptic transport and Weil-constant identity at a scalar norm
AutomorphicForm.exists_ellipticTransport_coupled_and_weilConst_mul_eq_of_not_isSigmaConjugate_scalar_of_finrank_eq_two36 below · cited by 1 · depth 31 - Finite class-sorted family spanning admissible section restrictions
AutomorphicForm.exists_fin_admissible_forall_flat_restrict_eq_sum_sameClass_of_principalLevel_archCutSubmodule29 below · cited by 1 · depth 31 - Finitely many ratios carry a non-zero window product
AutomorphicForm.exists_finset_forall_window_product_eq_zero_of_not_mem14 below · cited by 1 · depth 31 - Finite weight window for continuous cut vectors at a real place
AutomorphicForm.exists_forall_eq_sum_hasArchCharacterAtZero_mem_span_rightTranslate_of_mem_archCutSubmodule6 below · cited by 1 · depth 31 - Finite circle-weight window of a cut vector at a complex place
AutomorphicForm.exists_forall_eq_sum_hasCircleWeightAt_mem_span_rightTranslate_of_mem_archCutSubmodule_of_isArchSmoothAtComplex7 below · cited by 1 · depth 31 - Entire Euler-normalised non-constant part of adelic GL₂ Eisenstein series
AutomorphicForm.exists_forall_exists_entire_eulerProduct_mul_eq_bruhatEisenstein_sub_constantTerm_norm_le_mul_pow_archParam_weight_mul_rpow_neg_of_isCompact_of_flat138 below · cited by 1 · depth 31 - Uniform polynomial bound for partial L-factors on the unitary axis
AutomorphicForm.exists_forall_exists_entire_mul_eulerProduct_eq_and_ne_zero_and_norm_le_mul_pow_archParam_weight_mul_norm_of_isInducedSection_principalLevel163 below · cited by 1 · depth 31 - Hyperbolic class sums as finite sums of twisted lattice sums
AutomorphicForm.exists_forall_finsum_mul_prod_zpow_neg_mul_ideleNorm_mul_integral_orbital_eq_sum_tsum_ite_of_smul_eq_map_partAt_of_ne_one72 below · cited by 1 · depth 31 - Maass–Selberg relations on the unitary axis, two character pairs
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_axis_continuation_mul_conj_eq_maassSelberg_or_twoTerm_or_cross_or_zero_two_pairs_slab_of_flat277 below · cited by 1 · depth 31 - Integrability of the truncated σ-twisted continuous spectral expansion
AutomorphicForm.exists_forall_integrable_prod_sum_rightConv_mul_axis_continuation_mul_conj_lambdaT_sigmaAdelicAct_symm_of_subset_iUnion_image_centreCutSiegelSet442 below · cited by 1 · depth 31 - Haar measures on centralisers of split elements c(z) diag(u,1)
AutomorphicForm.exists_forall_isHaarMeasure_centralizer_centralScalar_mul_diagUnits2_and_integral_eq_mul_integral_prod_of_ne_one2 below · cited by 1 · depth 31 - Weil integration formula on GL₂(A_K) along unipotent fibres
AutomorphicForm.exists_forall_lintegral_mul_apply_col_det_eq_mul_lintegral_lintegral_ideleNorm_inv_of_ae_lintegral_mul_unipotentGL2_eq_one14 below · cited by 1 · depth 31 - Unit translation into an ample centre-cut Siegel set
AutomorphicForm.exists_forall_mem_centreCutSiegelSet_globalPoints_mul_mem_centreCutSiegelSetAmple1 below · cited by 1 · depth 31 - Germ expansion of elliptic orbital integrals near a central element
AutomorphicForm.exists_forall_nhds_scalar_forall_isOrbitalIntegral_eq_add_mul_of_mem_localCentralizer_of_not_isSquare13 below · cited by 1 · depth 31 - Uniform sup bound on K for flat induced families
AutomorphicForm.exists_forall_norm_apply_le_and_norm_axis_intertwining_apply_le_of_mem_adelicMaximalCompact_of_flat284 below · cited by 1 · depth 31 - Rapid decay of the ξ-averaged twisted kernel minus its constant term
AutomorphicForm.exists_forall_norm_setIntegral_mul_finsum_borel_div_mem_sub_constantTerm_centralScalar_mul_le_inv_adelicHeight_pow70 below · cited by 1 · depth 31 - Continuous-spectrum Plancherel identity for R(f) on the truncation domain
AutomorphicForm.exists_forall_setIntegral_convOp_continuousProjection_mul_conj_continuousProjection_eq_mul_tsum_integral_sum_rightConv_mul_setIntegral_mul_conj_axis_continuation1,049 below · cited by 1 · depth 31 - Self-adjointness of Arthur truncation on a Siegel-covered fundamental domain
AutomorphicForm.exists_forall_setIntegral_lambdaT_mul_conj_eq_setIntegral_lambdaT_mul_conj_lambdaT_of_subset_iUnion_image_centreCutSiegelSet24 below · cited by 1 · depth 31 - Combined orbital bound for Hecke double-coset test functions
AutomorphicForm.exists_forall_sum_lintegral_orbital_add_weightedOrbital_doubleCoset_le_mul_prod_rpow_measure273 below · cited by 1 · depth 31 - Integral generator Y and depth-m elements c(1+varpi^m Y)
AutomorphicForm.exists_generator_and_seq_mem_localCentralizer_tendsto_scalar_of_forall_not_diagonal0 below · cited by 2 · depth 31 - Twisted centraliser compact modulo real scalars when c<0
AutomorphicForm.exists_isCompact_forall_twistedCentralizer_conjAe_eq_scalar_mul_of_neg0 below · cited by 2 · depth 31 - Weyl symmetry of the local twisted-norm conditions
AutomorphicForm.exists_isNormOf_glArch_centralScalar_mul_diagUnits2_iff_inv_and_finComponent_iff_inv0 below · cited by 2 · depth 31 - Residue of the twisted-centralizer zeta integral for standard test functions
AutomorphicForm.exists_isOpen_isCompact_tendsto_sub_one_mul_lintegral_twistedCentralizer_schwartzMap_mul_indicator_nhdsGT_one_of_forall_integral_eq_mul_prod_integral_of_finrank_eq_two371 below · cited by 1 · depth 31 - Double-coset test function from its unit-coset translate
AutomorphicForm.exists_isSemiLocalFactorization_indicator_doubleCoset_of_isSemiLocalFactorization_indicator_translate0 below · cited by 1 · depth 31 - Existence of continuous twisted section functions on GL₂(L⊗_K A)
AutomorphicForm.exists_isTwistedSectionFnOn_and_continuous_of_isRegularSemisimple_normString_of_hasCompactSupport_of_isArtinianRing2 below · cited by 3 · depth 31 - Continuous twisted section functions over a locally compact field
AutomorphicForm.exists_isTwistedSectionFnOn_and_continuous_of_normString_eq_toTensorGL_scalar_of_algHom3 below · cited by 1 · depth 31 - Continuous twisted sections at real scalars over ℂ/ℝ
AutomorphicForm.exists_isTwistedSectionFnOn_conjAe_toTensorGL_scalar_and_continuous8 below · cited by 1 · depth 31 - Conjugation equivariance of local orbital integrals on GL₂(Kᵥ)
AutomorphicForm.exists_map_val_eq_map_conj_and_isOrbitalIntegral_conj_iff0 below · cited by 1 · depth 31 - A Borel unipotent coordinate on M₂(K_∞)
AutomorphicForm.exists_measurable_forall_apply_mul_unipotentGL2_eq_add_infiniteAdeleRing0 below · cited by 1 · depth 31 - Existence of a Bruhat function for the unipotent subgroup of GL₂(A_K)
AutomorphicForm.exists_measurable_forall_lintegral_mul_unipotentGL2_eq_one3 below · cited by 2 · depth 31 - Prime-degree twisted conjugacy: exact norm in the centraliser
AutomorphicForm.exists_mem_centralizer_normString_eq_toTensorGL_of_isNormOf_of_prime0 below · cited by 2 · depth 31 - Diagonal families covering hyperbolic classes realise and separate ratios
AutomorphicForm.exists_mem_ratio_eq_or_eq_inv_and_ratio_ne_of_ne_of_classReps0 below · cited by 1 · depth 31 - Complex place: K-finite induced vectors are polynomials in the bottom row
AutomorphicForm.exists_mvPolynomial_apply_eq_mul_eval_bottomRow_of_rightTranslatesSpanFinite_of_isComplex2 below · cited by 1 · depth 31 - Polynomiality in the bottom row at a real place
AutomorphicForm.exists_mvPolynomial_apply_eq_mul_eval_bottomRow_of_rightTranslatesSpanFinite_of_isReal1 below · cited by 1 · depth 31 - A regular norm pair on a prescribed valuation shell
AutomorphicForm.exists_ne_and_normString_diagUnits2_eq_toTensorGL_and_norm_eq_pow_inertiaDeg_mul_of_ramificationIdx_eq_one_of_prime3 below · cited by 1 · depth 31 - Uniform properness of twisted conjugation near a negative scalar norm
AutomorphicForm.exists_nhds_isCompact_forall_twistedCentralizer_conjAe_mul_mem_of_neg1 below · cited by 2 · depth 31 - Uniform compactness modulo twisted centraliser near the identity
AutomorphicForm.exists_nhds_isCompact_forall_twistedCentralizer_sigmaConj_mul_mem_of_not_isSigmaConjugate_scalar_of_finrank_eq_two1 below · cited by 1 · depth 31 - Germ of split orbital integrals near a central element
AutomorphicForm.exists_nhds_scalar_forall_isOrbitalIntegral_eq_mul_integral_unipotentGL2_conj_of_diagonal5 below · cited by 1 · depth 31 - Orbital integrals at regular diagonal elements of GL₂(ℂ)
AutomorphicForm.exists_pos_forall_isOrbitalIntegralOn_diagonal_complex_eq_mul_integral_unitaryAverage6 below · cited by 1 · depth 31 - Vanishing of the truncated constant-term defect integral
AutomorphicForm.exists_pos_forall_setIntegral_sub_constantTerm_mul_eq_zero_inter_lt_adelicHeight_of_subset_iUnion_image_centreCutSiegelSet24 below · cited by 1 · depth 31 - Godement sections realising a polynomial on the unit sphere
AutomorphicForm.exists_sum_mul_localZeta_line_polynomial_mul_gaussian_eq_eval3 below · cited by 1 · depth 31 - Twisted torus family along lifts of a split hyperbolic family
AutomorphicForm.exists_twistedTorusFamily_lift_centralScalar_mul_diagUnits2_coupled_massOne_restrictedProduct17 below · cited by 1 · depth 31 - Euler block times entire remainder in flat GL₂ Whittaker coefficients
AutomorphicForm.exists_whittakerCoefficient_diagOne_eq_eulerProduct_mul_entire_of_flat_family_of_unitary72 below · cited by 1 · depth 31 - One winding datum for all Hecke words (window side)
AutomorphicForm.exists_windingDatum_forall_heckeWord_mul_sum_slotFamilyCoeff_mul_sum_windowClassIntegral_eq_sum_satakeLaurent_mul_coeff302 below · cited by 1 · depth 31 - Entry-chart derivative at a complex place in invariant directions
AutomorphicForm.fderiv_apply_mul_archComplexLiftAt_eq_of_isArchSmoothAtComplex0 below · cited by 1 · depth 31 - Entry-chart derivative at a real place via H, E, F₋, centre
AutomorphicForm.fderiv_apply_mul_archRealLiftAt_eq_of_isArchSmoothAt0 below · cited by 1 · depth 31 - Finiteness modulo the centre and continuity of the folded kernel
AutomorphicForm.finite_setOf_exists_apply_globalPoints_out_mul_centralScalar_mul_ne_zero_and_continuous_finsum_integral_of_hasCompactSupport3 below · cited by 1 · depth 31 - Window cancellation at a non-norm idele, prime degree
AutomorphicForm.finrank_mul_ratio_mul_weightedClassIntegral_add_mul_window_eq_zero_of_mem_sup_of_not_mem_range_of_prime339 below · cited by 1 · depth 31 - Non-normic split classes: vanishing of the weighted window combination
AutomorphicForm.finrank_mul_ratio_mul_weightedClassIntegral_add_mul_window_eq_zero_of_ratio_not_mem_range_norm_of_prime344 below · cited by 1 · depth 31 - Twisted diagonal sum equals the σ-twisted cut trace
AutomorphicForm.finsum_setIntegral_convOp_mul_conj_sigmaAdelicAct_symm_eq_twistedCutTrace_of_orthonormal_principalLevel_of_isFundamentalDomain_slab367 below · cited by 1 · depth 31 - Archimedean derivations at distinct infinite places commute
AutomorphicForm.foldr_archDeriv_comm_of_ne_place0 below · cited by 2 · depth 31 - Regularity and slab bounds inherited by row-isometry translates
AutomorphicForm.forall_continuous_isArchSmoothAt_bounded_foldr_archDeriv_rightTranslate_rowIsometryInclAt6 below · cited by 1 · depth 31 - Regularity inherited by row-isometry translates at a complex place
AutomorphicForm.forall_continuous_isArchSmoothAt_bounded_foldr_archDeriv_rightTranslate_rowIsometryInclAt_of_isComplex6 below · cited by 2 · depth 31 - Pinned fine expansion of the truncated twisted hyperbolic term
AutomorphicForm.forall_exists_setIntegral_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_mul_sum_torusShellConst_mul_orbital_add_sum_weightedOrbital_of_isFactorizableTestFn210 below · cited by 2 · depth 31 - Truncated cuspidal kernel integrated along a Galois-twisted diagonal
AutomorphicForm.forall_integrableOn_and_setIntegral_lambdaT_tsum_convOp_mul_conj_sigmaAdelicAct_symm_eq_tsum_finsum_setIntegral_of_orthonormal_principalLevel_of_isFundamentalDomain_slab508 below · cited by 1 · depth 31 - Vanishing of the unipotent germ at a non-σ-conjugate scalar
AutomorphicForm.germ_unipotent_eq_zero_of_areMatchingLocal_of_forall_germ_of_not_isSigmaConjugate_scalar_of_finrank_eq_two3 below · cited by 1 · depth 31 - Raising and lowering operators shift the archimedean weight by 2
AutomorphicForm.hasArchCharacterAtZero_raise_lower_E_sub_Fm_of_hasArchCharacterAtZero_of_isArchSmoothAt2 below · cited by 1 · depth 31 - Circle-weight shifts of partial, partial̄ and H at a complex place
AutomorphicForm.hasCircleWeightAt_archDelAt_archDelBarAt_archDerivAtComplex_of_hasCircleWeightAt_of_isArchSmoothAtComplex0 below · cited by 1 · depth 31 - Expansion of int_A R(f)u along an orthonormal cusp system
AutomorphicForm.hasSum_setIntegral_mul_conj_mul_setIntegral_convOp_of_orthonormal_isotypicCuspSubmodule62 below · cited by 1 · depth 31 - Constant term of the centre-folded twisted GL₂ kernel
AutomorphicForm.integrableOn_and_measurable_and_constantTerm_setIntegral_mul_finsum_borel_div_mem_eq_setIntegral_mul_constantTerm_of_norm_ne_one7 below · cited by 1 · depth 31 - Unfolding a partial twisted class sum against the idele centre
AutomorphicForm.integrableOn_and_setIntegral_mul_finsum_sigmaConjClassOrbit_cosetFamily_eq_tsum_subtype_integral1 below · cited by 1 · depth 31 - Integrability of the S-part orbital-integral window at diag(u,1)
AutomorphicForm.integrable_mul_orbital_mul_prod_orbital_sPart_of_isArchTestFactor_of_isLocalTestFn32 below · cited by 1 · depth 31 - Adjointness of the GL₂ intertwining operator on the unitary axis
AutomorphicForm.integral_mul_conj_axis_continuation_weylIntertwiningIntegral_eq_of_swap_pair_of_principalLevel_of_ne_bot148 below · cited by 8 · depth 31 - Centre-integrated Euler factorisation of a hyperbolic class integral
AutomorphicForm.integral_mul_orbital_centralScalar_eq_mul_ideleNorm_mul_prod_tsum_mul_integral_of_isUnitFactorization_of_integrable23 below · cited by 1 · depth 31 - Archimedean twisted descent near a real scalar in GL₂
AutomorphicForm.isClosedEmbedding_toTensorGL_and_exists_nhds_one_twistedCentralizer_iff_and_forall_isCompact_conjAe4 below · cited by 2 · depth 31 - Left Casimirs at a complex place preserve level and types
AutomorphicForm.isFactorizableTestFn_leftCasimirComplex_and_rightConv_mem_of_isArchBiFinite_principal12 below · cited by 1 · depth 31 - Left Casimir of an admissible test function at principal level
AutomorphicForm.isFactorizableTestFn_leftCasimir_and_rightConv_mem_of_isArchBiFinite_principal11 below · cited by 1 · depth 31 - Factorisation against adelic Haar forces archimedean Haar and c_G>0
AutomorphicForm.isHaarMeasure_and_pos_of_forall_integral_adelicGLHaar_eq_mul_integral_mul_prod0 below · cited by 3 · depth 31 - Admissibility of the continued Weyl intertwining integral on the axis
AutomorphicForm.isInducedSection_and_continuous_and_isArchKFinite_axis_continuation_weylIntertwiningIntegral_of_forall_mem_principalLevel3 below · cited by 12 · depth 31 - Galois transport of a flat induced family and its continuation
AutomorphicForm.isInducedSection_and_isArchKFinite_and_axis_continuation_comp_sigmaAdelicAct_symm_of_flat_family_of_principalLevel7 below · cited by 2 · depth 31 - Unimodularity of the local twisted centralizer of a scalar-norm δ
AutomorphicForm.isInvInvariant_twistedCentralizer_completion_of_isNormConjugator_scalar2 below · cited by 1 · depth 31 - Translated indicators of GL₂(mathcal Oᵥ) and its principal congruence subgroup are local test functions
AutomorphicForm.isLocalTestFn_indicator_scalar_mul_localIntegralSet_and_indicator_principalCongruence0 below · cited by 3 · depth 31 - Local constancy and integrability of the split-family window product
AutomorphicForm.isLocallyConstant_finprod_unitValue_and_integrable_window_product_of_ne_one_of_isLocalTestFn60 below · cited by 1 · depth 31 - Orbital integrals of cK₀ and cK(𝔭) indicators at depth m
AutomorphicForm.isOrbitalIntegral_indicator_scalar_localIntegralSet_and_principalCongruence_of_depth_of_forall_not_diagonal20 below · cited by 1 · depth 31 - Archimedean twisted descent for GL₂(ℂ)/GL₂(ℝ)
AutomorphicForm.isTwistedOrbitalIntegralOn_conjAe_toTensorGL_mul_scalar_iff_isOrbitalIntegralOn_of_forall_integral_eq_one7 below · cited by 1 · depth 31 - Height-weighted orbital integrals at diagonal classes: linearity and measurability
AutomorphicForm.isWeightedOrbitalIntegralOn_sum_mul_centralScalar_mul_and_measurable_of_diagonal6 below · cited by 1 · depth 31 - Absolute convergence of the twisted-centralizer zeta integral for s₁>1
AutomorphicForm.lintegral_twistedCentralizer_enorm_mul_ideleNorm_det_rpow_lt_top_of_mem_schwartzBruhat2_of_forall_ne_scalar_of_finrank_eq_two323 below · cited by 1 · depth 31 - Mass 8π s of the Iwasawa box
AutomorphicForm.map_val_iwasawaBox_eq_of_gram_conjAe2 below · cited by 1 · depth 31 - Measurability of window values in the central idele parameter
AutomorphicForm.measurable_window_values_of_ne_one_of_prime59 below · cited by 1 · depth 31 - Positive finite measure for the compact torus in Z(γ₀)
AutomorphicForm.measure_setOf_mem_localCentralizer_pos_and_lt_top_of_forall_not_diagonal0 below · cited by 2 · depth 31 - Local L² property of automorphic forms on GL₂(A)
AutomorphicForm.memLp_two_restrict_of_isCompact_of_isAutomorphicFnAt_canonicalTruncationDomain26 below · cited by 7 · depth 31 - Diagonal units of an integral upper-triangular matrix lie in R
AutomorphicForm.mem_and_inv_mem_of_upperTriangular_mem_integralUnitsSet0 below · cited by 2 · depth 31 - Residual members are spanned by continuous characters χ∘det
AutomorphicForm.mem_span_chiDet_continuous_of_mem_residualSpan_of_isAutomorphicFnAt25 below · cited by 4 · depth 31 - Orbital integral at a scalar in GL₂(Kᵥ)
AutomorphicForm.mul_measure_localIntegralSet_eq_apply_of_isOrbitalIntegral_scalar0 below · cited by 1 · depth 31 - Vanishing of the ∞–S orbital window from class vanishing
AutomorphicForm.mul_prod_orbital_eq_zero_of_forall_apply_conj_centralScalar_mul_diagUnits2_eq_zero6 below · cited by 1 · depth 31 - Orbital integrals of spherical Hecke words at split regular elements
AutomorphicForm.norm_sub_mul_absNorm_pow_mul_eq_T_add_T_neg_one_pow_apply_of_isOrbitalIntegral_sum_indicator_heckeWord_diagonal15 below · cited by 2 · depth 31 - Weil constant times band mass equals shell mass
AutomorphicForm.ofReal_mul_inf_twistedCentralizer_detBand_eq_twistedCentralizer_detShell_of_weilConst0 below · cited by 1 · depth 31 - Translates of derivative words lie in spans of equal-length words
AutomorphicForm.rightTranslate_foldr_archDeriv_mem_span_foldr_archDeriv_rightTranslate_of_isComplex7 below · cited by 1 · depth 31 - Semi-local evaluation intertwines the idèlic Galois action with σ⊗ 1
AutomorphicForm.semiLocalEval_act_eq_congr_and_semiLocalIdele_unitsAct_and_semiLocalComponent_sigmaAdelicAct1 below · cited by 6 · depth 31 - Peel step for the semi-local central transfer
AutomorphicForm.semilocal_central_transfer_peel_step6 below · cited by 1 · depth 31 - Peel step for semi-local central transfer with reference measures
AutomorphicForm.semilocal_central_transfer_referenceMeasures_peel_step6 below · cited by 1 · depth 31 - Centre unfolding of the truncated hyperbolic term over K
AutomorphicForm.setIntegral_canonicalTruncationDomain_adelicKernelHyperbolicPart_sub_indicator_constantTerm_eq_mul_sum_mul_integral_add_sum_of_eq_mul_sum_orbital_add_sum_weightedOrbital56 below · cited by 1 · depth 31 - Twisted hyperbolic term via orbital integrals over norm-one ideles
AutomorphicForm.setIntegral_canonicalTruncationDomain_finsum_hyperbolicCell_sub_indicator_constantTerm_eq_mul_sum_mul_integral_haarQuotient_ker_idelicNorm_add_sum_of_eq_mul_sum_orbital_add_sum_weightedOrbital74 below · cited by 1 · depth 31 - Orthogonality of χ∘det on the canonical truncation domain
AutomorphicForm.setIntegral_chiDet_mul_conj_chiDet_canonicalTruncationDomain_eq_and_eq_zero_of_ne_of_squaresToXi19 below · cited by 3 · depth 31 - Right translation by unit determinant norm preserves fundamental-domain integrals
AutomorphicForm.setIntegral_comp_mul_eq_setIntegral_of_isFundamentalDomain_of_ideleNorm_det_eq_one8 below · cited by 13 · depth 31 - Continuous part: integral over A as normalised pairing with u^Aₑ
AutomorphicForm.setIntegral_convOp_continuousProjection_eq_inv_mul_setIntegral_convOp_mul_conj_continuousProjection88 below · cited by 1 · depth 31 - Unfolding an automorphisation against a continuous ξ-equivariant function
AutomorphicForm.setIntegral_finsum_integral_indicator_mul_conj_eq_mul_setIntegral_mul_conj_of_continuous_of_isLsXiFunction72 below · cited by 3 · depth 31 - Unfolding an automorphised test function against an automorphic function
AutomorphicForm.setIntegral_finsum_integral_indicator_mul_conj_eq_mul_setIntegral_mul_conj_of_isAutomorphicFnAt60 below · cited by 4 · depth 31 - Cuspidal component of the automorphised indicator
AutomorphicForm.setIntegral_mul_conj_eq_mul_setIntegral_inter_conj_of_lsXi_threeWay_of_mem_isotypicCuspSubmodule74 below · cited by 1 · depth 31 - Cusp forms are orthogonal to the residual span
AutomorphicForm.setIntegral_mul_conj_eq_zero_of_ae_constantTerm_eq_zero_of_mem_residualSpan_slab12 below · cited by 6 · depth 31 - Fubini for the Eisenstein kernel on a measurable rectangle
AutomorphicForm.setIntegral_prod_tsum_integral_sum_rightConv_axis_continuation_mul_conj_eq_tsum_integral_sum_mul_setIntegral_indicator_mul_conj440 below · cited by 1 · depth 31 - Subtraction-free theta decomposition of a twisted-centralizer zeta integral
AutomorphicForm.setLIntegral_mul_ideleNorm_det_rpow_add_eq_setLIntegral_reflectPair_add_lintegral_mul_rate_of_isFundamentalDomain_twistedCentralizer_of_forall_ne_scalar_of_finrank_eq_two48 below · cited by 1 · depth 31 - Unfolding the GL₂ theta integral over a fundamental domain
AutomorphicForm.setLIntegral_mul_tsum_apply_mulVec_eq_mul_measure_mul_lintegral_mul_setLIntegral_of_isFundamentalDomain_op7 below · cited by 1 · depth 31 - Archimedean constant of the Gram-normalised measure on GL₂
AutomorphicForm.setLIntegral_ofReal_norm_det_eq_mul_two_pow_mul_two_pi_pow_of_map_coe_eq_smul_withDensity_gram_infiniteAdeleRing8 below · cited by 1 · depth 31 - σ-action preserves the adelic maximal compact and its Haar measure
AutomorphicForm.sigmaAdelicAct_mem_adelicMaximalCompact_and_integral_maximalCompactHaar_comp_sigmaAdelicAct4 below · cited by 2 · depth 31 - Left invariance of Hecke word counts under central units
AutomorphicForm.sum_indicator_localIntegralSet_prod_mul_zpow_inv_mul_centralUnit_mul0 below · cited by 2 · depth 31 - Double-coset volume bound for twisted orbital class sums
AutomorphicForm.sum_lintegral_orbital_add_weightedOrbital_indicator_translate_mul_prod_measure_doubleCoset_le18 below · cited by 1 · depth 31 - Weighted intercept comparison for base change on GL(2)
AutomorphicForm.sum_mul_integral_haarQuotient_ker_idelicNorm_weighted_eq_finrank_mul_slopeFactor_mul_sum_sum_add_window_of_forall_eq_add1 below · cited by 1 · depth 31 - Local Hecke slot combination at a split shell class
AutomorphicForm.sum_slotCoeff_mul_tsum_pow_mul_eq_inv_norm_sub_one_mul_ite_of_isOrbitalIntegral_heckeWord_diagonal_zpow17 below · cited by 2 · depth 31 - Slot regrouping of Hecke-word orbital integrals
AutomorphicForm.sum_slotFamilyCoeff_mul_prod_eq_prod_of_isOrbitalIntegral_heckeWord_of_isOrbitalIntegral_sum_coeff_univWord6 below · cited by 1 · depth 31 - The torus-shell constant κ₀ evaluated
AutomorphicForm.torusShell_const_eq_of_forall_lintegral_eq31 below · cited by 1 · depth 31 - Twisted cuspidal trace under norm twist and level change
AutomorphicForm.tsum_twistedCutTrace_eq_tsum_twistedCutTrace_principalLevel_mul_ideleNorm_det_rpow_of_isFundamentalDomain_slab36 below · cited by 1 · depth 31 - Quaternionic twisted centralizer: determinant shell has mass 32π² s
AutomorphicForm.twistedCentralizer_detShell_eq_of_gram_conjAe_of_neg2 below · cited by 1 · depth 31 - Per-class window transfer for twisted weighted orbital integrals
AutomorphicForm.twistedWeightedClassIntegral_eq_finrank_mul_ratio_mul_weightedClassIntegral_add_mul_window_of_coupled_of_isSemiLocalFactorization76 below · cited by 1 · depth 31 - Weighted base-change identity J'=[L:K] J at an unramified place
AutomorphicForm.twistedWeighted_eq_finrank_mul_weighted_heckeWord_of_unramified55 below · cited by 1 · depth 31 - Weighted fundamental lemma at an unramified place: J'=[L:K]J
AutomorphicForm.twistedWeighted_eq_finrank_mul_weighted_indicator_of_unramified20 below · cited by 1 · depth 31 - Meromorphic continuation of the Weyl intertwining integral, flat families
AutomorphicForm.weylIntertwiningIntegral_meromorphicOn_of_flat_family74 below · cited by 1 · depth 31 - Invariance of window values under (u,z)↦(u⁻¹,zu)
AutomorphicForm.window_values_inv_mul_unitsMap_eq_of_ne_one_of_prime26 below · cited by 1 · depth 31 - Galois invariance of the adelic height on GL₂
AutomorphicForm.adelicHeight_sigmaAdelicAct6 below · cited by 2 · depth 32 - Diagonal invariance and continuity of the archimedean weight
AutomorphicForm.archWeight_archIdentGL_diagonal_mul_and_continuous0 below · cited by 4 · depth 32 - Continued Weyl intertwining integral preserves archimedean types
AutomorphicForm.axis_continuation_weylIntertwiningIntegral_mem_archCutSubmodule_of_forall_mem_archCutSubmodule2 below · cited by 7 · depth 32 - Finite component at v of c(z) diag(a,b)
AutomorphicForm.coe_finComponent_glFin_centralScalar_mul_diagUnits20 below · cited by 14 · depth 32 - Smoothing upgrades almost-everywhere cuspidality to pointwise vanishing
AutomorphicForm.constantTerm_convOp_eq_zero_of_ae_constantTerm_eq_zero_of_isAutomorphicFnAt27 below · cited by 2 · depth 32 - Hecke translate of an orthogonal cuspidal remainder vanishes
AutomorphicForm.convOp_ae_eq_zero_restrict_canonicalTruncationDomain_of_ae_constantTerm_eq_zero_of_forall_setIntegral_mul_conj_eq_zero57 below · cited by 1 · depth 32 - Elliptic orbital integral on GL₂(Kᵥ) in an affine chart
AutomorphicForm.eq_div_mul_integral_norm_inv_smul_conj_affineChart_of_isOrbitalIntegral_of_not_isSquare2 below · cited by 1 · depth 32 - Archimedean orbital integrals scale inversely with centraliser Haar measure
AutomorphicForm.eq_inv_mul_of_isOrbitalIntegralOn_of_isOrbitalIntegralOn_smul_infiniteAdeleRing2 below · cited by 3 · depth 32 - Archimedean twisted weighted orbital integrals: lift independence and scaling
AutomorphicForm.eq_inv_mul_of_isTwistedWeightedOrbitalIntegralOn_of_normString_eq_toTensorGL_diagonal_of_coupled_one_smul_infiniteAdeleRing5 below · cited by 4 · depth 32 - Twisted weighted orbital integral of the unit at a split place
AutomorphicForm.eq_ite_finrank_mul_sum_of_isTwistedWeightedOrbitalIntegral_indicator_semiLocalIntegralSet_of_nontrivial_extension10 below · cited by 1 · depth 32 - Twisted weighted orbital integral of the unit at an inert place
AutomorphicForm.eq_ite_finrank_mul_sum_of_isTwistedWeightedOrbitalIntegral_indicator_semiLocalIntegralSet_of_subsingleton_extension14 below · cited by 1 · depth 32 - Unit orbital integral at the split class diag(au,a)
AutomorphicForm.eq_ite_inv_norm_sub_one_of_isOrbitalIntegral_indicator_localIntegralSet_diagonal4 below · cited by 5 · depth 32 - Weighted orbital integral of the spherical unit at diag(a,b)
AutomorphicForm.eq_ite_sum_of_isWeightedOrbitalIntegral_indicator_localIntegralSet_diagUnits24 below · cited by 2 · depth 32 - Regular diagonal orbital integral descends to the unipotent radical
AutomorphicForm.eq_norm_inv_mul_integral_localIntegralSet_integral_conj_unipotentGL2_of_isOrbitalIntegral_of_diagonal3 below · cited by 5 · depth 32 - Uniqueness of the central–elliptic comparison constant
AutomorphicForm.eq_of_forall_setIntegral_centralElliptic_eq_mul_sum_of_exists_areMatchingAt_sum_ne_zero0 below · cited by 1 · depth 32 - Orbital integral independent of normalised Haar measure on the centraliser
AutomorphicForm.eq_of_isOrbitalIntegral_of_isOrbitalIntegral_of_measure_preimage_localIntegralSet_eq_one4 below · cited by 3 · depth 32 - Local orbital integrals agree under central-unit translation of a diagonal class
AutomorphicForm.eq_of_isOrbitalIntegral_of_isOrbitalIntegral_smul_diagonal_of_forall_centralUnit_mul4 below · cited by 2 · depth 32 - Lift independence of local twisted weighted orbital integrals
AutomorphicForm.eq_of_isTwistedWeightedOrbitalIntegral_of_isTwistedWeightedOrbitalIntegral_of_normString_eq_toTensorGL_diagonal4 below · cited by 8 · depth 32 - Quaternionic twisted centraliser measure in an ℝ⁴ chart
AutomorphicForm.exists_chart_map_val_eq_smul_withDensity_of_gram_conjAe_of_neg0 below · cited by 1 · depth 32 - Archimedean window for split-torus orbital integrals on GL₂
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_units_prod_norm_sub_one_pow_mul_eq_of_isOrbitalIntegralOn_glArch_centralScalar_mul_diagUnits220 below · cited by 3 · depth 32 - Uniform smooth archimedean window for split orbital integrals
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_units_prod_norm_sub_one_pow_mul_eq_of_isOrbitalIntegralOn_glArch_centralScalar_mul_diagUnits2_of_ne_one20 below · cited by 4 · depth 32 - Smooth archimedean window for normalised split orbital integrals
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_units_prod_norm_sub_one_pow_mul_eq_of_isOrbitalIntegralOn_scalar_mul_diagUnits217 below · cited by 4 · depth 32 - Row isometries at a real place form O(2)
AutomorphicForm.exists_continuousMulEquiv_rowIsometrySubgroup_orthogonalGroup_of_isReal0 below · cited by 2 · depth 32 - Row isometries of GL₂(F_w) at a complex place form U(2)
AutomorphicForm.exists_continuousMulEquiv_rowIsometrySubgroup_unitaryGroup_of_isComplex0 below · cited by 2 · depth 32 - One continuous compactly supported window for twisted archimedean orbital integrals
AutomorphicForm.exists_continuous_hasCompactSupport_eq_of_isTwistedWeightedOrbitalIntegralOn_glArch_centralScalar_mul_diagUnits216 below · cited by 3 · depth 32 - A single archimedean window for weighted split orbital integrals
AutomorphicForm.exists_continuous_hasCompactSupport_tsupport_subset_units_eq_of_isWeightedOrbitalIntegralOn_glArch_centralScalar_mul_diagUnits220 below · cited by 4 · depth 32 - Integral norm-string lift of diagonal units at unramified places
AutomorphicForm.exists_diagUnits2_mem_semiLocalIntegralSet_and_normString_eq_toTensorGL_of_ramificationIdx_eq_one7 below · cited by 1 · depth 32 - Semi-local double coset: finite disjoint coset decomposition and volume
AutomorphicForm.exists_doubleCoset_semiLocalIntegralSet_eq_iUnion_smul_and_semiLocalHaar_eq_card0 below · cited by 3 · depth 32 - Elliptic transport with coupled measures at a finite place
AutomorphicForm.exists_ellipticTransport_coupled_straighten_of_not_isSigmaConjugate_scalar_of_finrank_eq_two6 below · cited by 1 · depth 32 - Continuous characters of the real row-isometry group are integral
AutomorphicForm.exists_eq_archWeightCharReal_of_continuous0 below · cited by 1 · depth 32 - Diagonality of lifts with regular split diagonal norm string
AutomorphicForm.exists_eq_diagUnits2_of_normString_eq_toTensorGL_diagUnits20 below · cited by 6 · depth 32 - Level-U coset formula for local orbital integrals
AutomorphicForm.exists_finset_isOrbitalIntegral_sum_mul_div_of_forall_mul_eq_of_isOpen2 below · cited by 1 · depth 32 - Level structure for a non-split twisted centralizer, [L:K]=2
AutomorphicForm.exists_finset_level_isOpen_isCompact_box_subset_indicator_mulVec_eq_prod_indicator_tensorPlace_of_normString_eq_toTensorGL_centralScalar_of_finrank_eq_two293 below · cited by 2 · depth 32 - Bad-place set of a non-norm idelic class in GL₂
AutomorphicForm.exists_finset_not_isNormOf_and_not_card_eq_one_of_mem_sup_of_not_mem_range_of_prime286 below · cited by 1 · depth 32 - Non-normic diagonal ratio: the bad place set is no singleton
AutomorphicForm.exists_finset_not_isNormOf_and_not_card_eq_one_of_ratio_not_mem_range_norm_of_prime291 below · cited by 1 · depth 32 - Counting hyperbolic σ-classes contributing to a double-coset orbital integral
AutomorphicForm.exists_forall_card_le_mul_prod_pow_log_measure_of_lintegral_orbital_doubleCoset_ne_zero24 below · cited by 1 · depth 32 - L² boundedness of right convolution on the truncation domain
AutomorphicForm.exists_forall_eLpNorm_convOp_le_mul_eLpNorm_restrict_canonicalTruncationDomain_of_isAutomorphicFnAt34 below · cited by 2 · depth 32 - Uniform Euler factorisation and decay of Eisenstein Whittaker coefficients
AutomorphicForm.exists_forall_exists_entire_whittakerCoefficient_bruhatEisenstein_eq_eulerProduct_mul_norm_tsum_le_mul_pow_archParam_weight_mul_rpow_neg_of_isCompact_of_flat117 below · cited by 1 · depth 32 - Uniform dimension bound for induced sections restricted to K
AutomorphicForm.exists_forall_exists_submodule_maximalCompact_finrank_le_restrict_mem_of_isInducedSection_principalLevel_archCutSubmodule7 below · cited by 1 · depth 32 - Maass–Selberg relations on the unitary axis for flat sections
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_axis_continuation_mul_conj_eq_maassSelberg_or_twoTerm_or_cross_or_zero_two_pairs_canonicalTruncationDomain_of_flat274 below · cited by 1 · depth 32 - Truncated inner products independent of the truncation datum
AutomorphicForm.exists_forall_integrableOn_iff_and_setIntegral_lambdaT_mul_conj_lambdaT_eq_of_isTruncationDatum_of_isTruncationDatum11 below · cited by 1 · depth 32 - Uniqueness of the invariant measure on the orbit of (e₁,1)
AutomorphicForm.exists_forall_lintegral_eq_mul_lintegral_mul_apply_col_det_of_forall_map_mulVec_eq_self3 below · cited by 1 · depth 32 - Haar measure on an elliptic torus in (p,r)-coordinates
AutomorphicForm.exists_forall_lintegral_localCentralizer_eq_mul_lintegral_prod_norm_inv_of_not_isSquare2 below · cited by 1 · depth 32 - Haar measure on GL₂(Kᵥ) in matrix coordinates
AutomorphicForm.exists_forall_lintegral_localHaar_eq_mul_lintegral_pi_norm_det_inv_sq2 below · cited by 1 · depth 32 - Per-class double-coset bound for twisted orbital integrals
AutomorphicForm.exists_forall_lintegral_orbital_doubleCoset_le_mul_prod_rpow_measure248 below · cited by 1 · depth 32 - Logarithmic weight bound for twisted orbital double-coset integrals
AutomorphicForm.exists_forall_lintegral_weightedOrbital_doubleCoset_le_mul_prod_pow_log_measure_mul_lintegral_orbital27 below · cited by 1 · depth 32 - Paley–Wiener spectral form of R(f) on the continuous spectrum
AutomorphicForm.exists_forall_setIntegral_convOp_continuousProjection_pseudoEisenstein_mul_conj_eq_mul_tsum_integral_sum_rightConv_axis_pairing_of_matched_paleyWiener520 below · cited by 1 · depth 32 - A universal homogeneous germ α for elliptic orbital integrals
AutomorphicForm.exists_forall_sq_mul_eq_norm_mul_and_forall_integral_affineChart_eq_add_mul_apply_one_of_not_isSquare2 below · cited by 1 · depth 32 - Twisted convolution carries isotypic cut blocks to a transported block
AutomorphicForm.exists_forall_twistedConvOp_mem_isotypicCuspSubmodule_inf_archCutSubmodule_principalLevel_of_isBiInvariantUnder_of_isFundamentalDomain_slab31 below · cited by 3 · depth 32 - Adelic lift with prescribed semi-local component at one place
AutomorphicForm.exists_glArch_eq_one_and_semiLocalComponent_glFin_eq_of_mem_semiLocalIntegralSet0 below · cited by 1 · depth 32 - L² limits of automorphic functions; cuspidality closed and linear
AutomorphicForm.exists_isAutomorphicFnAt_ae_eq_of_tendsto_eLpNorm_and_ae_constantTerm_eq_zero_canonicalTruncationDomain23 below · cited by 2 · depth 32 - Non-vanishing orbital integrals confine the idele to a compact set
AutomorphicForm.exists_isCompact_forall_mem_of_orbital_partAt_ne_zero_of_mem_unitIdelesOutside4 below · cited by 1 · depth 32 - Compact support of archimedean orbital values in the ratio a
AutomorphicForm.exists_isCompact_forall_ratio_mem_of_isOrbitalIntegralOn_infiniteAdeleRing_scalar_mul_diagUnits20 below · cited by 1 · depth 32 - Compactness of the ratio locus of non-vanishing orbital values
AutomorphicForm.exists_isCompact_forall_ratio_mem_of_isOrbitalIntegral_scalar_mul_diagUnits20 below · cited by 1 · depth 32 - Compactness of the a-support of archimedean twisted orbital values
AutomorphicForm.exists_isCompact_forall_ratio_mem_of_isTwistedWeightedOrbitalIntegralOn_infiniteAdeleRing_of_normString_eq_toTensorGL_scalar_mul_diagUnits20 below · cited by 1 · depth 32 - Compact support bound for twisted weighted orbital values
AutomorphicForm.exists_isCompact_forall_ratio_mem_of_isTwistedWeightedOrbitalIntegral_of_normString_eq_toTensorGL_scalar_mul_diagUnits20 below · cited by 1 · depth 32 - Non-vanishing archimedean weighted orbital values confine the ratio a
AutomorphicForm.exists_isCompact_forall_ratio_mem_of_isWeightedOrbitalIntegralOn_infiniteAdeleRing_scalar_mul_diagUnits20 below · cited by 1 · depth 32 - Compact support in the ratio a for weighted orbital values
AutomorphicForm.exists_isCompact_forall_ratio_mem_of_isWeightedOrbitalIntegral_scalar_mul_diagUnits20 below · cited by 1 · depth 32 - Haar measures on local twisted centralisers normalised on integral points
AutomorphicForm.exists_isHaarMeasure_twistedCentralizer_tensorPlace_preimage_semiLocalIntegralSet_eq_one0 below · cited by 2 · depth 32 - A locally constant compactly supported twisted weighted local window at v
AutomorphicForm.exists_isLocallyConstant_hasCompactSupport_eq_of_isTwistedWeightedOrbitalIntegral_finComponent_glFin_centralScalar_mul_diagUnits210 below · cited by 3 · depth 32 - Local weighted window of the split torus family at a finite place
AutomorphicForm.exists_isLocallyConstant_hasCompactSupport_eq_of_isWeightedOrbitalIntegral_finComponent_glFin_centralScalar_mul_diagUnits27 below · cited by 3 · depth 32 - Finite-place window function for a family of orbital integrals
AutomorphicForm.exists_isLocallyConstant_hasCompactSupport_norm_sub_one_mul_eq_of_isOrbitalIntegral_finComponent_glFin_centralScalar_mul_diagUnits27 below · cited by 4 · depth 32 - Normalised split orbital integrals as a test function on the torus
AutomorphicForm.exists_isLocallyConstant_hasCompactSupport_norm_sub_one_mul_eq_of_isOrbitalIntegral_scalar_mul_diagUnits25 below · cited by 3 · depth 32 - Split regular diagonal is a σ-norm iff both entries are norms
AutomorphicForm.exists_isNormOf_diagUnits2_iff_mem_range_norm_of_isUnit_sub2 below · cited by 3 · depth 32 - Semi-local factorisation with word indicators at T
AutomorphicForm.exists_isSemiLocalFactorization_word2 below · cited by 1 · depth 32 - Leibniz expansion of a twisted weighted orbital integral
AutomorphicForm.exists_isTwistedWeightedOrbitalIntegralOn_baseChange_eq_mul_sum_prod_of_isSemiLocalFactorization9 below · cited by 1 · depth 32 - Convergence of the twisted-centralizer zeta integral for s₁>1
AutomorphicForm.exists_lintegral_twistedCentralizer_inv_one_add_norm_sq_pow_mul_indicator_mul_ideleNorm_det_rpow_lt_top_of_forall_ne_scalar_of_finrank_eq_two319 below · cited by 1 · depth 32 - Archimedean zeta integral of bihomogeneous Gaussian as a Γ_ℝ-factor
AutomorphicForm.exists_localZeta_line_eq_mul_GammaReal_mul_of_bihomogeneous_mul_gaussian2 below · cited by 1 · depth 32 - Split regular elements near a non-norm scalar are no norms
AutomorphicForm.exists_nhds_forall_not_exists_isNormOf_diagonal_of_not_isSigmaConjugate_scalar_of_finrank_eq_two0 below · cited by 1 · depth 32 - Uniform properness of twisted conjugation near real scalars
AutomorphicForm.exists_nhds_one_forall_isCompact_exists_eq_toTensorGL_mul_of_conjAe_twistedConj_mem2 below · cited by 1 · depth 32 - Unipotent orbital integral: Iwasawa versus affine-chart normalisation
AutomorphicForm.exists_pos_forall_integral_localIntegralSet_integral_unipotentGL2_conj_eq_mul_integral_affineChart4 below · cited by 1 · depth 32 - Coupled Haar measures on a product split factorwise
AutomorphicForm.exists_prod_eq_and_coupled_of_coupled_prod0 below · cited by 2 · depth 32 - Local test functions are bi-invariant under an open subgroup
AutomorphicForm.exists_subgroup_isOpen_subset_localIntegralSet_forall_mul_eq_of_isLocalTestFn0 below · cited by 2 · depth 32 - Prescribing components of GL₂(L⊗_KA_K)
AutomorphicForm.exists_tensorArch_eq_and_forall_tensorPlace_eq_of_forall_not_mem_mem_semiLocalIntegralSet_and_ext0 below · cited by 1 · depth 32 - Euler expansion of a weighted adelic orbital integral at a diagonal class
AutomorphicForm.exists_weightedClassIntegral_eq_mul_archWindow_mul_prod_add_mul_sum_window_and_isWeightedOrbitalIntegral_of_isUnitFactorization_of_coupled50 below · cited by 3 · depth 32 - Winding-datum realisation of the unweighted window class sum
AutomorphicForm.exists_windingDatum_forall_coeff_eq_window_classSum_of_areMatchingArch_of_areMatchingLocal_of_ne_one_unweighted255 below · cited by 1 · depth 32 - Vanishing of the combined weighted Euler bracket
AutomorphicForm.finrank_mul_ratio_mul_add_mul_window_eq_zero_of_forall_mem_eq_zero_of_not_singleton0 below · cited by 2 · depth 32 - Moving the σ-twist across the convolution in block pairings
AutomorphicForm.finsum_setIntegral_sigmaSectionActOn_convOp_mul_conj_eq_finsum_setIntegral_twistedConvOp_mul_conj_of_orthonormal_principalLevel_of_isFundamentalDomain_slab366 below · cited by 1 · depth 32 - Continuous-spectrum form of R(f) for automorphised bounded cutoffs
AutomorphicForm.forall_setIntegral_convOp_continuousProjection_mul_conj_eq_mul_tsum_integral_sum_rightConv_axis_pairing_of_forall_paleyWiener1,015 below · cited by 1 · depth 32 - Invariance of ground window values under (u,z)↦(u⁻¹,zι u)
AutomorphicForm.ground_window_values_inv_mul_unitsMap_eq_of_ne_one15 below · cited by 1 · depth 32 - Central fold of a twisted GL₂ kernel: convergence and Fubini
AutomorphicForm.integrableOn_and_measurable_and_constantTerm_setIntegral_mul_finsum_borel_div_mem_eq_setIntegral_mul_integral_finsum_inv_unipotentGL2_mul3 below · cited by 1 · depth 32 - Trace-class convolution on the cut cuspidal spectrum, Galois-twisted
AutomorphicForm.integrableOn_convOp_mul_conj_sigmaAdelicAct_symm_and_summable_setIntegral_norm_finsum_of_orthonormal_principalLevel_of_isFundamentalDomain_slab113 below · cited by 1 · depth 32 - Change of fundamental domain in a determinant slab
AutomorphicForm.integrableOn_iff_and_setIntegral_eq_and_setIntegral_comp_sigmaAdelicAct_symm_eq_of_invariant_of_isFundamentalDomain_slab8 below · cited by 4 · depth 32 - Unipotent translation invariance of box averages of twisted GL₂ sums
AutomorphicForm.integrable_and_integral_finsum_borel_div_mem_inv_unipotentGL2_mul_eq_integral_finsum_of_norm_ne_one4 below · cited by 1 · depth 32 - Centre unfolding of hyperbolic orbital integrals over K
AutomorphicForm.integral_haarQuotient_orbital_eq_const_mul_integral_of_isOrbitalIntegralOn_centralScalar_mul55 below · cited by 1 · depth 32 - H-quotient versus norm-one twisted orbital integrals for GL₂
AutomorphicForm.integral_haarQuotient_twistedOrbital_eq_const_mul_integral_quotient_ker_idelicNorm_of_isTwistedOrbitalIntegralOn73 below · cited by 2 · depth 32 - Adjointness of the Weyl intertwining integral for Re s>1/2
AutomorphicForm.integral_mul_conj_weylIntertwiningIntegral_eq_integral_weylIntertwiningIntegral_mul_conj_of_re_gt_half19 below · cited by 1 · depth 32 - Twisted orbital integrals over a product of groups
AutomorphicForm.integral_twistedConj_prod_mul_eq_mul_integral_integral_of_sigmaCentralizer1 below · cited by 2 · depth 32 - Right convolution preserves automorphy at the truncation-domain pins
AutomorphicForm.isAutomorphicFnAt_convOp_of_isAutomorphicFnAt_canonicalTruncationDomain34 below · cited by 8 · depth 32 - Twisted centralizers of near-central elements in GL₂(ℂ⊗_ℝℝ)
AutomorphicForm.isClosedEmbedding_toTensorGL_and_exists_nhds_one_mem_twistedCentralizer_conjAe_iff0 below · cited by 1 · depth 32 - Twisting isotypic cusp forms by ‖det‖^{w/2} between levels U₁(N) and U(N)
AutomorphicForm.isIsotypicCuspFormAt_mul_ideleNorm_det_rpow_principalLevel_and_levelOne_of_isFundamentalDomain_slab10 below · cited by 1 · depth 32 - Central translation between test function and twisted class, weighted case
AutomorphicForm.isTwistedWeightedOrbitalIntegralOn_comp_scalar_mul_iff0 below · cited by 2 · depth 32 - Archimedean unipotent fibre integration for GL₂
AutomorphicForm.lintegral_mul_apply_col_det_eq_mul_lintegral_setLIntegral_of_map_coe_eq_smul_withDensity_gram_infiniteAdeleRing5 below · cited by 1 · depth 32 - Left translation by an integral element fixes twisted orbital class terms
AutomorphicForm.lintegral_orbital_comp_inv_mul_eq_and_weightedOrbital_eq_of_glArch_eq_one_of_isSemiLocalFactorization15 below · cited by 1 · depth 32 - Gram-normalised measure on GL₂(ℝ) and its Iwasawa box
AutomorphicForm.map_entries_eq_smul_withDensity_and_apply_iwasawaBox_eq_of_gram_real1 below · cited by 1 · depth 32 - Positive torus unit volume and finite twisted-centralizer unit volume
AutomorphicForm.measureReal_torusUnits_pos_and_measure_detUnits_lt_top_of_not_isSigmaConjugate_scalar_of_finrank_eq_two4 below · cited by 1 · depth 32 - Cartan double coset membership in GL₂(Kᵥ) via norms
AutomorphicForm.mem_localIntegralSet_mul_singleton_diagonal_mul_localIntegralSet_iff_norm0 below · cited by 5 · depth 32 - Orbit vanishing forces a product of local orbital integrals to vanish
AutomorphicForm.mul_prod_eq_zero_of_forall_apply_conj_eq_zero_of_isUnitFactorization0 below · cited by 1 · depth 32 - Norm string of a diagonal matrix in base-changed GL₂
AutomorphicForm.normString_apply_eq_one_tmul_norm_apply_of_diagonal1 below · cited by 4 · depth 32 - Adjointness of right convolution on the truncation domain
AutomorphicForm.setIntegral_mul_conj_convOp_eq_setIntegral_convOp_conj_inv_mul_conj_of_isAutomorphicFnAt31 below · cited by 3 · depth 32 - Central–elliptic comparison for cyclic base change, constant displayed
AutomorphicForm.setIntegral_twistedEllipticCentralFold_eq_const_mul_sum_of_factorization_of_normFibre907 below · cited by 1 · depth 32 - Covolume identity for the test function g⊗mathbf 1_U
AutomorphicForm.sqrt_det_gram_mul_lintegral_schwartzMap_archIdent_mul_prod_corr_eq_lintegral_pairHaar_mul_two_pow_mul_discr_sq_of_isOpen_isCompact94 below · cited by 1 · depth 32 - Cancellation of the weighted terms outside S_K
AutomorphicForm.sub_finrank_mul_ratio_mul_eq_mul_window_arch_add_sum_window_of_forall_eq_of_forall_eq_finrank_mul0 below · cited by 1 · depth 32 - Unweighted window class sums as a winding pairing
AutomorphicForm.sum_slotFamilyCoeff_mul_sum_mul_integral_window_eq_sum_prod_mul_windingDatum_coeff_of_forall_coeff_eq_of_ne_one_unweighted100 below · cited by 1 · depth 32 - Euler limit of the twisted-centralizer zeta integral at a level
AutomorphicForm.tendsto_sub_one_mul_lintegral_twistedCentralizer_schwartzMap_mul_indicator_nhdsGT_one_of_level_of_covol28 below · cited by 1 · depth 32 - Integrality descends along GL₂(Kᵥ)toGL₂(L⊗_K Kᵥ)
AutomorphicForm.toTensorGL_mem_semiLocalIntegralSet_iff_mem_localIntegralSet0 below · cited by 2 · depth 32 - Poisson summation over a σ-twisted centralizer in GL₂
AutomorphicForm.tsum_sigmaCentralizer_apply_mulVec_add_eq_inv_ideleNorm_det_mul_tsum_reflectPair_of_forall_ne_scalar_of_finrank_eq_two39 below · cited by 1 · depth 32 - Twisted centraliser of a norm-exact diagonal element
AutomorphicForm.twistedCentralizer_diagUnits2_eq_map_toTensorGL_centralizer_of_normString_eq_of_isUnit_sub3 below · cited by 6 · depth 32 - Central translates do not change the twisted centraliser
AutomorphicForm.twistedCentralizer_scalar_mul0 below · cited by 4 · depth 32 - Weighted fundamental lemma for Hecke words at an inert place
AutomorphicForm.twistedWeighted_eq_finrank_mul_weighted_heckeWord_of_inertiaDeg_eq_finrank37 below · cited by 1 · depth 32 - Weighted Hecke-word fundamental lemma at a split place
AutomorphicForm.twistedWeighted_eq_finrank_mul_weighted_heckeWord_of_inertiaDeg_eq_one27 below · cited by 1 · depth 32 - Invariance of twisted window values under (u,z)↦(u⁻¹,zι u)
AutomorphicForm.twisted_window_values_inv_mul_unitsMap_eq_of_ne_one_of_prime15 below · cited by 1 · depth 32 - Local and semi-local weights are invariant under (twisted) centralisers of diagonal elements
AutomorphicForm.weight_localCentralizer_mul_and_semiLocalWeight_twistedCentralizer_mul_of_diagonal0 below · cited by 4 · depth 32 - Weil constant times unit volume of the elliptic torus
AutomorphicForm.weilConst_mul_measureReal_torusUnits_eq_of_isNormConjugator_of_not_isSigmaConjugate_scalar_of_finrank_eq_two26 below · cited by 1 · depth 32 - Cuspidality as orthogonality to all pseudo-Eisenstein series
AutomorphicForm.ae_constantTerm_eq_zero_iff_forall_setIntegral_pseudoEisenstein_mul_conj_eq_zero_slab65 below · cited by 4 · depth 33 - Cuspidal part vanishes; residual part is the residual projection
AutomorphicForm.ae_eq_zero_cuspidalPart_of_forall_isSlabProfile_setIntegral_pseudoEisenstein_mul_conj_eq_zero22 below · cited by 2 · depth 33 - Invariance, continuity and measurability of the archimedean height weight
AutomorphicForm.archWeight_centralizer_mul_and_continuous_and_aestronglyMeasurable_of_diagonal0 below · cited by 3 · depth 33 - Stability of the archimedean cut under determinant-one compact translation
AutomorphicForm.comp_mul_mem_archCutSubmodule_of_mem_adelicMaximalCompact_of_det_archComponent_eq_one0 below · cited by 3 · depth 33 - Axis pairing of (φ,R(f)ψ) as Eisenstein-coefficient sum
AutomorphicForm.conj_sum_integral_axis_pairing_add_eq_mul_tsum_integral_sum_rightConv_mul_thetaPairing_of_matched_paleyWiener363 below · cited by 1 · depth 33 - Unfolding a central-translate orbital integral over H_Kbackslash GL₂(mathbb A_K)
AutomorphicForm.const_mul_eq_integral_haarQuotient_centralScalar_of_isOrbitalIntegralOn_of_diagonal52 below · cited by 1 · depth 33 - Centre-unfolding identity for twisted orbital integrals on GL₂
AutomorphicForm.const_mul_eq_integral_haarQuotient_integral_ker_idelicNorm_centralScalar_of_isTwistedOrbitalIntegralOn_comp_baseChangeGL24 below · cited by 1 · depth 33 - Determinants on the twisted centralizer are scalar; unit-determinant part is compact open
AutomorphicForm.det_mem_range_and_isOpen_isCompact_detUnits_twistedCentralizer_of_not_isSigmaConjugate_scalar_of_finrank_eq_two2 below · cited by 5 · depth 33 - Archimedean weighted orbital values scale inversely with torus measure
AutomorphicForm.eq_inv_mul_of_isWeightedOrbitalIntegralOn_of_isWeightedOrbitalIntegralOn_smul_infiniteAdeleRing2 below · cited by 2 · depth 33 - Twisted weighted orbital integral of the unit at an inert place
AutomorphicForm.eq_ite_finrank_mul_sum_of_isTwistedWeightedOrbitalIntegral_indicator_semiLocalIntegralSet_of_relIndex_eq_of_subsingleton_extension7 below · cited by 1 · depth 33 - Uniqueness of local weighted orbital integrals at diagonal elements
AutomorphicForm.eq_of_isWeightedOrbitalIntegral_of_isWeightedOrbitalIntegral_diagonal_of_measure_preimage_localIntegralSet_eq_one3 below · cited by 6 · depth 33 - Norm of the eigenvalue ratio separates hyperbolic σ-classes
AutomorphicForm.eq_of_norm_div_eq_norm_div_of_mem_of_disjoint_sigmaClasses4 below · cited by 1 · depth 33 - K-side value of a weighted word orbital integral
AutomorphicForm.eq_two_mul_log_mul_shellValue_of_isWeightedOrbitalIntegral_baseChange_heckeWord14 below · cited by 1 · depth 33 - Weighted orbital integral at a split diagonal element as a shell sum
AutomorphicForm.eq_two_mul_log_mul_sum_of_isWeightedOrbitalIntegral_diagUnits2_of_biInvariant4 below · cited by 2 · depth 33 - Transport of semi-local GL₂ data at an inert place
AutomorphicForm.exists_algEquiv_mulEquiv_semiLocalComponent_localEmbed_eq_of_subsingleton_extension1 below · cited by 1 · depth 33 - Smooth archimedean window for unipotent orbital integrals
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_units_prod_norm_pow_mul_integral_integral_unipotentGL2_eq_of_isArchTestFactor6 below · cited by 1 · depth 33 - Local twisted commutant: column map is a topological isomorphism
AutomorphicForm.exists_continuousLinearEquiv_twistedCommutant_tensor_adicCompletion_mulVec_of_forall_ne_scalar3 below · cited by 1 · depth 33 - Weighted archimedean orbital integrals along central translates of a split class
AutomorphicForm.exists_continuous_hasCompactSupport_eq_of_isWeightedOrbitalIntegralOn_scalar_mul_diagUnits217 below · cited by 1 · depth 33 - Sign–central–determinant-one factorisation in the adelic maximal compact
AutomorphicForm.exists_diagOne_sign_mul_centralScalar_mul_eq_of_mem_adelicMaximalCompact0 below · cited by 3 · depth 33 - Integrality of a column detects integrality in the local twisted commutant at almost all places
AutomorphicForm.exists_finset_forall_mem_twistedCommutant_tensor_adicCompletion_mulVec_mem_semiLocalIntegers_iff_of_forall_ne_scalar3 below · cited by 1 · depth 33 - Height weight bounded on the support of twisted double-coset integrands
AutomorphicForm.exists_forall_abs_log_adelicHeight_mul_adelicHeight_adelicWeyl_le_mul_prod_pow_log_measure_of_doubleCoset_apply_ne_zero26 below · cited by 1 · depth 33 - Uniform dilation bound for Whittaker coefficients of flat Eisenstein families
AutomorphicForm.exists_forall_exists_whittakerCoefficient_diagOne_eq_eulerProduct_mul_entire_norm_le_mul_pow_archParam_weight_dilation_of_flat107 below · cited by 1 · depth 33 - Shalika germ expansion at the identity for GL₂
AutomorphicForm.exists_forall_integral_affineChart_eq_add_mul_apply_one_of_not_isSquare0 below · cited by 1 · depth 33 - Uniform bound on orthonormal systems of adelic induced sections
AutomorphicForm.exists_forall_le_of_orthonormal_maximalCompact_isInducedSection_principalLevel_archCutSubmodule_of_ne_bot2 below · cited by 1 · depth 33 - Product decomposition of Haar measure on the adelic maximal compact
AutomorphicForm.exists_forall_lintegral_and_integral_maximalCompactHaar_eq_mul_prod_semiLocalHaar2 below · cited by 1 · depth 33 - Polynomial growth on the unitary axis of the continued intertwining integral
AutomorphicForm.exists_forall_norm_axis_continuation_weylIntertwiningIntegral_le_mul_pow_of_flat165 below · cited by 3 · depth 33 - Unit-box bounds for contributing twisted hyperbolic classes
AutomorphicForm.exists_forall_norm_div_mem_unitBox_of_lintegral_orbital_doubleCoset_ne_zero16 below · cited by 1 · depth 33 - One-place double-coset bound for twisted orbital integrals
AutomorphicForm.exists_forall_norm_sub_norm_mul_le_mul_rpow_mul_log_pow_of_isTwistedOrbitalIntegral_indicator_doubleCoset35 below · cited by 1 · depth 33 - Uniform bound for twisted orbital integrals at regular diagonal elements
AutomorphicForm.exists_forall_norm_sub_norm_mul_le_of_isTwistedOrbitalIntegral_of_isSemiLocalTestFn46 below · cited by 1 · depth 33 - Archimedean twisted orbital bound, uniform in central translates
AutomorphicForm.exists_forall_prod_infinitePlace_norm_sub_norm_mul_le_of_isTwistedOrbitalIntegralOn_tensorArch_scalar_mul35 below · cited by 1 · depth 33 - Maass–Selberg relations on the unitary axis, two character pairs
AutomorphicForm.exists_forall_setIntegral_lambdaT_axis_continuation_mul_conj_eq_maassSelberg_and_eq_twoTerm_and_eq_cross_and_eq_zero_two_pairs_slab_of_ne268 below · cited by 1 · depth 33 - Eisenstein coefficients of a matched pseudo-Eisenstein series on the unitary axis
AutomorphicForm.exists_forall_setIntegral_pseudoEisenstein_mul_conj_axis_continuation_eq_mul_integral_mul_conj_add_integral_mul_conj_weylIntertwining_of_matched_paleyWiener86 below · cited by 5 · depth 33 - Intercept class sums as lattice sums of kink windows
AutomorphicForm.exists_forall_window_classSum_eq_tsum_mul_tsum_ite_kinkWindow_of_areMatchingArch_of_areMatchingLocal_of_ne_one_unweighted237 below · cited by 1 · depth 33 - Determinant norms on the support lie in a fixed compact translate
AutomorphicForm.exists_isCompact_forall_idelicNorm_det_mul_mem_of_isSemiLocalFactorization_indicator_doubleCoset25 below · cited by 1 · depth 33 - Compactness of the b-locus for archimedean twisted orbital integrals
AutomorphicForm.exists_isCompact_forall_mem_of_isTwistedWeightedOrbitalIntegralOn_infiniteAdeleRing_of_normString_eq_toTensorGL_scalar_mul_diagUnits20 below · cited by 1 · depth 33 - Compactness of the split parameter for nonvanishing twisted weighted orbital integrals
AutomorphicForm.exists_isCompact_forall_mem_of_isTwistedWeightedOrbitalIntegral_of_normString_eq_toTensorGL_scalar_mul_diagUnits20 below · cited by 1 · depth 33 - Haar measure on an archimedean twisted centraliser via K_∞^×× K_∞^×
AutomorphicForm.exists_isHaarMeasure_twistedCentralizer_tensorArch_integral_eq_integral_prod_toTensorGL_diagUnits20 below · cited by 1 · depth 33 - Euler factorisation of central-translate orbital and height-weighted orbital integrals
AutomorphicForm.exists_isOrbitalIntegralOn_centralScalar_mul_eq_mul_prod_and_isWeightedOrbitalIntegralOn_eq_mul_sum_prod_of_isUnitFactorization11 below · cited by 1 · depth 33 - Central translation preserves semi-local factorisation, after enlarging S
AutomorphicForm.exists_isSemiLocalFactorization_comp_centralScalar_mul1 below · cited by 1 · depth 33 - Weighted cyclic-string integral equals (n+1)J for one spherical factor
AutomorphicForm.exists_isWeightedOrbitalIntegral_and_integral_pi_mul_prod_indicator_cyclicString_eq_mul_of_forall_mul_eq6 below · cited by 1 · depth 33 - A K-basis for the twisted commutant of δ₀
AutomorphicForm.exists_linearIndependent_forall_mul_eq_mul_map_iff_mem_span_of_normString_eq_toTensorGL_centralScalar_of_forall_ne_scalar3 below · cited by 2 · depth 33 - Matched Paley–Wiener pair approximating two automorphisations jointly
AutomorphicForm.exists_matched_paleyWiener_pair_forall_norm_setIntegral_sub_le_and_tsum_integral_sum_normSq_sub_setIntegral_axis_continuation_le1,006 below · cited by 1 · depth 33 - Galois twist matched by a cut cuspidal vector
AutomorphicForm.exists_mem_iSup_isotypicCuspSubmodule_inf_archCutSubmodule_convOp_sigmaSectionActOn_eq_and_setIntegral_mul_conj_eq_principalLevel_of_isFundamentalDomain_slab363 below · cited by 1 · depth 33 - A twist of δ with regular elliptic rational norm
AutomorphicForm.exists_mem_twistedCentralizer_isRegularSemisimple_not_isSquare_isNormOf_mul_of_not_isSigmaConjugate_scalar_of_finrank_eq_two1 below · cited by 1 · depth 33 - Uniformiser determinant in a twisted centralizer of GL₂
AutomorphicForm.exists_mem_twistedCentralizer_valuation_eq_exp_neg_one_and_det_eq_of_isNormOf_scalar_of_finrank_eq_two21 below · cited by 2 · depth 33 - Non-split twisted class: δ₀σ(δ₀) central and twisted commutant a division algebra
AutomorphicForm.exists_mul_map_eq_scalar_and_forall_isUnit_of_normString_eq_toTensorGL_centralScalar_of_forall_ne_scalar2 below · cited by 10 · depth 33 - Central transport of twisted weighted orbital integral values
AutomorphicForm.exists_nhds_forall_exists_isTwistedWeightedOrbitalIntegral_of_normString_eq_toTensorGL_scalar_mul_diagUnits23 below · cited by 2 · depth 33 - Norm sweep near a scalar along a fixed torus
AutomorphicForm.exists_nhds_forall_mem_localCentralizer_isNormConjugator_mul_of_isNormConjugator_mul1 below · cited by 1 · depth 33 - Archimedean twisted weighted orbital integrals along a central direction
AutomorphicForm.exists_nhds_forall_pow_eq_and_continuousOn_and_eq_of_isTwistedWeightedOrbitalIntegralOn_comp_toTensorGL_scalar_mul_infiniteAdeleRing4 below · cited by 1 · depth 33 - Norms of conjugate products at an unramified place
AutomorphicForm.exists_norm_eq_zpow_and_norm_eq_zpow_of_prod_algEquiv_pow_eq0 below · cited by 1 · depth 33 - Norm of 1-ba⁻¹ is a power of N(v)⁻¹
AutomorphicForm.exists_norm_one_sub_mul_inv_eq_zpow_neg_of_ne0 below · cited by 1 · depth 33 - Base change pushes Haar on GL₂(L⊗_KA_K) to adelic Haar
AutomorphicForm.exists_pos_forall_integral_comp_baseChangeGL_eq_mul_integral_adelicGLHaar0 below · cited by 1 · depth 33 - Haar integration on the σ-twisted diagonal centraliser of GL₂(A_L)
AutomorphicForm.exists_pos_forall_integral_sigmaCentraliser_eq_mul_integral_prod_centralScalar_mul_baseChangeGL_diagUnits24 below · cited by 1 · depth 33 - Class-uniform constant in the twisted torus factorisation
AutomorphicForm.exists_pos_forall_integral_twistedCentralizer_eq_mul_integral_tensorArch_mul_prod_integral_tensorPlace_of_diagonal1 below · cited by 1 · depth 33 - Archimedean column map bounded below on the twisted commutant
AutomorphicForm.exists_pos_forall_norm_le_mul_norm_archIdent_sum_smul_mulVec_tmul_of_linearIndependent_of_span_eq5 below · cited by 2 · depth 33 - Archimedean descent of split orbital integrals to the torus
AutomorphicForm.exists_pos_forall_prod_norm_sub_one_pow_mul_eq_mul_prod_norm_pow_mul_integral_integral_of_isOrbitalIntegralOn_scalar_mul_diagUnits214 below · cited by 1 · depth 33 - Prescribing finitely many local components of an adelic matrix
AutomorphicForm.exists_tensorArch_eq_and_forall_tensorPlace_eq_of_finset0 below · cited by 1 · depth 33 - Hilbert 90 for L⊗_K K_∞, cyclic case
AutomorphicForm.exists_units_eq_sigmaTensor_mul_inv_of_prod_iterate_sigmaTensor_eq_one_infiniteAdeleRing1 below · cited by 2 · depth 33 - Idelic norm of det(c(w)cdotbc(x⁻¹δ ^σ x))
AutomorphicForm.idelicNorm_det_centralScalar_mul_baseChangeGL_inv_mul_mul_sigmaGL0 below · cited by 1 · depth 33 - Integrability of a twisted orbital integral at a regular class
AutomorphicForm.integrable_integral_character_mul_twistedOrbital_haarQuotient_of_norm_ne_one_of_trivial_on_principal73 below · cited by 1 · depth 33 - Integrability of the window bracket against the S-part measure
AutomorphicForm.integrable_mul_window_bracket_sPart_of_isWeightedOrbitalIntegralOn_of_isTwistedWeightedOrbitalIntegralOn_of_ne_one59 below · cited by 1 · depth 33 - Fubini over the centre for a regular diagonal class
AutomorphicForm.integral_haarQuotient_integral_character_mul_orbital_eq_integral_character_mul_integral_haarQuotient_centralScalar51 below · cited by 1 · depth 33 - Folding a twisted central integral over the norm-one ideles
AutomorphicForm.integral_haarQuotient_integral_character_mul_twistedOrbital_eq_integral_quotient_ker_idelicNorm_character_mul_integral_haarQuotient_integral54 below · cited by 1 · depth 33 - Twisted weighted word orbital integral at an inert place
AutomorphicForm.integral_heckeWord_twistedConj_mul_weight_eq_two_mul_log_mul_twistedShellValue24 below · cited by 1 · depth 33 - Shift-twisted weighted orbital integral of the unit equals (n+1)c
AutomorphicForm.integral_indicator_shiftTwistedConj_mul_sum_weight_mul_eq_mul_of_forall_isWeightedOrbitalIntegral_eq4 below · cited by 1 · depth 33 - Twisted weighted orbital integral equals a unipotent integral
AutomorphicForm.integral_twistedConj_map_algEquiv_mul_weight_eq_integral_unipotentGL2_of_biInvariant5 below · cited by 2 · depth 33 - Pseudo-Eisenstein series of a slab profile is automorphic
AutomorphicForm.isAutomorphicFnAt_pseudoEisenstein_slab22 below · cited by 20 · depth 33 - Compact, relatively open conjugated integral twisted commutant
AutomorphicForm.isCompact_and_exists_isOpen_conj_integralOrder_twistedCommutant_of_map_conj_eq_smul_map_toTensorGL_localHaar0 below · cited by 1 · depth 33 - Compact open maximal order in a twisted commutant
AutomorphicForm.isCompact_and_exists_isOpen_maximalOrder_twistedCommutant_of_not_isSigmaConjugate_scalar_of_finrank_eq_two3 below · cited by 2 · depth 33 - Galois twist of an isotypic cusp form at principal level
AutomorphicForm.isIsotypicCuspFormAt_sigmaSectionActOn_principalLevel_of_isFundamentalDomain_slab14 below · cited by 1 · depth 33 - Local constancy and compact support of semi-local box functions
AutomorphicForm.isLocallyConstant_and_hasCompactSupport_indicator_prod_semiLocalEval0 below · cited by 2 · depth 33 - Compact open adelic level set from prescribed semi-local data
AutomorphicForm.isOpen_and_isCompact_and_nonempty_and_exists_box_subset_of_forall_semiLocalEval_mem1 below · cited by 1 · depth 33 - Central scalar translation of local orbital integrals
AutomorphicForm.isOrbitalIntegral_scalar_mul_and_isWeightedOrbitalIntegral_scalar_mul_of_comp_scalar_mul0 below · cited by 1 · depth 33 - Smoothing an a.e.-cuspidal automorphic function on GL₂
AutomorphicForm.isSmoothCuspAutomorphicFnAt_convOp_and_continuous_and_mem_archCutSubmodule_of_ae_constantTerm_eq_zero40 below · cited by 1 · depth 33 - Central translation for local twisted orbital integrals
AutomorphicForm.isTwistedOrbitalIntegral_scalar_mul_of_isTwistedOrbitalIntegral_comp_scalar_mul0 below · cited by 2 · depth 33 - Weyl invariance of the Iwasawa integral on GL₂(A_F)
AutomorphicForm.lintegral_lintegral_adelicWeyl_mul_unipotentGL2_mul_eq_of_forall_centralScalar_mul_diagOne_mul_eq10 below · cited by 1 · depth 33 - Lower integral equals real part of orbital double integral
AutomorphicForm.lintegral_lintegral_ofReal_norm_twistedOrbital_eq_ofReal_re_integral_integral_of_nonneg0 below · cited by 1 · depth 33 - Euler factorisation of a twisted-centralizer zeta integral
AutomorphicForm.lintegral_twistedCentralizer_mul_indicator_mul_ideleNorm_det_rpow_eq_mul_lintegral_arch_mul_dedekindZeta_mul_prod_of_forall_le_mul_one_add_norm_rpow_neg19 below · cited by 2 · depth 33 - Finiteness of the archimedean zeta integral over a twisted centraliser
AutomorphicForm.lintegral_twistedCentralizer_mul_rpow_abs_algebraNorm_det_lt_top_of_map_coe_eq_smul_withDensity_gram_of_forall_le_mul_one_add_norm_rpow_neg0 below · cited by 1 · depth 33 - Uniform normalisation and Weyl symmetry of archimedean torus measures
AutomorphicForm.map_subtypeVal_centralizer_eq_and_map_conj_adelicWeyl_eq_of_forall_integral_eq_mul_integral_prod3 below · cited by 1 · depth 33 - Push-forward of archimedean centraliser measures is independent of z
AutomorphicForm.map_subtypeVal_eq_map_subtypeVal_of_forall_integral_centralizer_eq_mul_integral_prod3 below · cited by 1 · depth 33 - Shell measures for the twisted difference θ y - cy
AutomorphicForm.measureReal_norm_eq_and_norm_algEquiv_sub_mul_eq_of_prod_eq_of_ramificationIdx_eq_one6 below · cited by 2 · depth 33 - Unit mass and Euler product for twisted centralizer integrals
AutomorphicForm.measure_semiLocalIntegralSet_eq_one_and_tendsto_prod_lintegral_twistedCentralizer_of_forall_integral_eq_mul_prod_integral1 below · cited by 2 · depth 33 - Annihilator criterion for membership in a sum of type pieces
AutomorphicForm.mem_iSup_typeSubmodule_iff_forall_finsupp_sum_smul_eq_zero0 below · cited by 6 · depth 33 - Any Kᵥ-splitting computes semi-local integers and weights
AutomorphicForm.mem_semiLocalIntegers_iff_forall_mem_and_semiLocalWeight_eq_sum_weight_of_algEquiv_pi0 below · cited by 1 · depth 33 - Twisted centraliser of a diagonal element with unit norm difference
AutomorphicForm.mem_twistedCentralizer_iff_of_diagonal_of_isUnit_norm_sub_norm2 below · cited by 2 · depth 33 - Real structure on L⊗_K K_∞: finiteness and continuity
AutomorphicForm.moduleFinite_and_continuousSMul_real_tensor_infiniteAdeleRing_and_continuous_algebraNorm_det0 below · cited by 8 · depth 33 - Local twisted zeta integral for a non-scalar σ-conjugacy class
AutomorphicForm.ne_top_and_setLIntegral_iUnion_detShell_twistedCentralizer_norm_algebraNorm_det_rpow_eq_mul_and_tendsto_of_not_isSigmaConjugate_scalar_of_finrank_eq_two26 below · cited by 2 · depth 33 - Local zeta integral over a twisted centraliser, first-kind normalisation
AutomorphicForm.ne_top_and_setLIntegral_twistedCentralizer_conj_integral_norm_algebraNorm_det_rpow_eq_mul_of_map_conj_eq_smul_map_toTensorGL_localHaar_of_finrank_eq_two3 below · cited by 1 · depth 33 - Finiteness of s in a Haar pushforward s·ν
AutomorphicForm.ne_top_of_isHaarMeasure_twistedCentralizer_of_map_coe_eq_smul0 below · cited by 2 · depth 33 - Norm of a twisted difference θ y - cy when ‖c‖ ≠ 1
AutomorphicForm.norm_algEquiv_sub_mul_eq_norm_mul_max_of_norm_ne_one0 below · cited by 2 · depth 33 - Stability of the R(f)-pairing under weak L² approximation
AutomorphicForm.norm_setIntegral_convOp_mul_conj_sub_le_of_forall_norm_setIntegral_sub_mul_conj_le44 below · cited by 1 · depth 33 - Central translation preserves a twisted orbital integral bound
AutomorphicForm.norm_sub_norm_mul_le_of_isTwistedOrbitalIntegral_comp_scalar_mul1 below · cited by 1 · depth 33 - Unit fundamental lemma inequality at an unramified place
AutomorphicForm.norm_sub_norm_mul_le_of_isTwistedOrbitalIntegral_indicator_semiLocalIntegralSet_of_ramificationIdx_eq_one29 below · cited by 1 · depth 33 - Entrywise norms of a base-change lift are global norms
AutomorphicForm.norm_tensorPlace_apply_eq_algebraMap_norm_of_baseChangeGL_eq_globalPoints0 below · cited by 2 · depth 33 - Stability of the Eisenstein coefficient form of R(f)
AutomorphicForm.norm_tsum_integral_sum_rightConv_mul_mul_conj_sub_le_of_tsum_integral_sum_normSq_sub_le12 below · cited by 1 · depth 33 - Finite-adelic covolume identity with local correction factors
AutomorphicForm.prod_corr_one_mul_sqrt_discr_pow_mul_norm_det_mul_measure_pi_adelicBox_eq_measure_colPreimage_mul_two_pow_mul_prod_mul_discr_sq78 below · cited by 1 · depth 33 - Lattice index at an inert unramified place of prime degree
AutomorphicForm.relIndex_semiLocalIntegers_comap_sigmaTensor_sub_mulLeft_eq_absNorm_pow_min_of_subsingleton_extension5 below · cited by 1 · depth 33 - Right translation by K¹ preserves flat induced families and L²-norm
AutomorphicForm.rightTranslate_adelicMaximalCompact_det_one_isInducedSection_isArchKFinite_isKfSmooth_flat_principalLevel_archCutSubmodule_and_integral_norm_sq_eq2 below · cited by 1 · depth 33 - Semi-local component above v of a twisted unipotent product
AutomorphicForm.semiLocalComponent_glFin_inv_mul_unipotentGL2_mul_diagOne_mul_centralScalar_mul_sigmaAdelicAct2 below · cited by 1 · depth 33 - Continuous-spectrum part of a pseudo-Eisenstein pairing as a difference
AutomorphicForm.setIntegral_continuousPart_mul_conj_convOp_continuousPart_eq_sub_of_pseudoEisenstein_threeWay81 below · cited by 1 · depth 33 - Axis Parseval identity for the pair (φ, R(f)ψ)
AutomorphicForm.setIntegral_pseudoEisenstein_mul_conj_convOp_pseudoEisenstein_sub_eq_mul_sum_integral_axis_pairing_convOp_of_paleyWiener351 below · cited by 1 · depth 33 - Weil's integration formula on GL₂(A) along unipotent fibres
AutomorphicForm.setLIntegral_mul_apply_col_det_mul_inv_norm_det_sq_eq_lintegral_setLIntegral_of_forall_lintegral_mul_unipotent_eq_one0 below · cited by 1 · depth 33 - Galois twist intertwines right convolution on GL₂(A_L)
AutomorphicForm.sigmaSectionActOn_convOp_and_twistedConvOp_eq_sigmaSectionActOn_convOp1 below · cited by 1 · depth 33 - Adelic covolume identity for the twisted-commutant column map
AutomorphicForm.sqrt_det_gram_mul_lintegral_schwartzMap_archIdent_mul_measure_colPreimage_eq_lintegral_pairHaar_mul_sqrt_discr_pow_mul_norm_det_mul_measure_pi_adelicBox19 below · cited by 1 · depth 33 - Conjugation exchanges the archimedean cut and dual cut
AutomorphicForm.star_mem_archCutSubmodule_and_star_mem_archDualCutSubmodule_of_continuous1 below · cited by 4 · depth 33 - Residue at s'=1 of a twisted adelic zeta integral
AutomorphicForm.tendsto_sub_one_mul_archFactor_mul_dedekindZeta_mul_prod_nhdsGT_one_of_map_coe_eq_smul_withDensity_gram_of_covolume5 below · cited by 1 · depth 33 - Twisted centralizer of a diagonal element with regular split norm
AutomorphicForm.twistedCentralizer_diagUnits2_eq_map_toTensorGL_centralizer_of_normString_eq_of_prime1 below · cited by 3 · depth 33 - Unit determinant forces integral trace in an elliptic torus
AutomorphicForm.valuation_trace_le_one_of_valuation_det_eq_one_of_mem_localCentralizer_of_forall_not_diagonal0 below · cited by 1 · depth 33 - Weil constant times torus unit volume, ramified and unramified cases
AutomorphicForm.weilConst_mul_measureReal_torusUnits_eq_of_detUnits_isCompact_isOpen_of_isNormConjugator2 below · cited by 1 · depth 33 - Locality of the window bracket in the S-and-infinity coordinates
AutomorphicForm.window_bracket_eq_window_bracket_partAt_of_isWeightedOrbitalIntegralOn_of_isTwistedWeightedOrbitalIntegralOn_of_ne_one21 below · cited by 1 · depth 33 - A.e. vanishing of the constant term is an L²-class invariant
AutomorphicForm.ae_constantTerm_eq_zero_of_ae_eq_restrict_slab12 below · cited by 1 · depth 34 - Convolution by an arch-type-bi-finite test function stays in the arch cut
AutomorphicForm.convOp_mem_archCutSubmodule_of_isArchBiFinite_of_isAutomorphicFnAt28 below · cited by 2 · depth 34 - Right convolution preserves the residual projection property
AutomorphicForm.convOp_residualProjection_of_residualProjection38 below · cited by 2 · depth 34 - Non-degeneracy of the trace form on the twisted commutant
AutomorphicForm.det_trace_matrix_trace_mul_ne_zero_of_forall_mul_eq_mul_map_iff_mem_span_of_normString_eq_toTensorGL_centralScalar3 below · cited by 1 · depth 34 - Young-type L² bound for right convolution on truncation domains
AutomorphicForm.eLpNorm_convOp_le_ofReal_integral_norm_mul_eLpNorm_restrict_canonicalTruncationDomain_of_isAutomorphicFnAt34 below · cited by 1 · depth 34 - Twisted-shift decomposition of L⊗_K Kᵥ for cyclic L/K
AutomorphicForm.exists_algEquiv_pi_adicCompletion_forall_sigmaTensor_apply_eq_of_forall_mem_zpowers3 below · cited by 1 · depth 34 - Archimedean discrepancy window for cyclic base change
AutomorphicForm.exists_contDiff_hasCompactSupport_archDisc_mul_twistedWeighted_sub_finrank_mul_weighted_eq_add_sum_real_add_sum_complex_of_isCompact90 below · cited by 2 · depth 34 - Archimedean discrepancy of twisted and standard weighted orbital integrals
AutomorphicForm.exists_contDiff_hasCompactSupport_forall_prod_norm_sub_one_pow_mul_twistedWeighted_sub_finrank_mul_weighted_eq_mul_archDisc_of_areMatchingArch94 below · cited by 1 · depth 34 - Continuous section functions for adelic twisted orbital integrals
AutomorphicForm.exists_continuous_isTwistedSectionFnOn_adeleRing_of_isRegularSemisimple_normString10 below · cited by 1 · depth 34 - Column-preimage mass equals adelic box mass times local masses
AutomorphicForm.exists_finset_measure_colPreimage_mul_prod_measure_pi_integers_eq_measure_pi_adelicBox_mul_prod_measure_preimage_level11 below · cited by 1 · depth 34 - Semi-local test functions dominated by finitely many double-coset indicators
AutomorphicForm.exists_finset_norm_le_mul_sum_indicator_semiLocalIntegralSet_mul_mul_of_isSemiLocalTestFn0 below · cited by 1 · depth 34 - Uniform Whittaker factorisation for flat level-N Eisenstein pieces
AutomorphicForm.exists_forall_exists_whittakerCoefficient_bruhatEisenstein_diagOne_eq_cpowChar_mul_sum_eulerProduct_norm_le_on_balls_and_of_re_mem_Icc_of_flat91 below · cited by 1 · depth 34 - Truncated inner products of GL₂ Eisenstein series: cross and vanishing cases
AutomorphicForm.exists_forall_integrableOn_and_setIntegral_lambdaT_pseudoEisenstein_mul_conj_eq_cross_and_eq_zero_two_pairs_slab_of_re_lt_re104 below · cited by 1 · depth 34 - Maximal compact Haar measure factorises over semi-local places
AutomorphicForm.exists_forall_lintegral_maximalCompactHaar_eq_lintegral_mul_prod_setLIntegral_semiLocalHaar1 below · cited by 1 · depth 34 - Uniform bound for archimedean twisted-orbital volumes
AutomorphicForm.exists_forall_lintegral_mul_indicator_mul_sigmaTensor_mul_inv_le_of_isCompact1 below · cited by 1 · depth 34 - Uniform normalisation of archimedean torus measures along split classes
AutomorphicForm.exists_forall_map_entries_centralizer_eq_of_forall_integral_centralScalar_mul_diagUnits2_eq_mul_integral_mul_prod1 below · cited by 2 · depth 34 - Support window for the Iwasawa ratios of a double-coset test function
AutomorphicForm.exists_forall_norm_div_window_of_doubleCoset_apply_borel_mul_maximalCompact_ne_zero8 below · cited by 2 · depth 34 - Uniform bound for twisted orbital integrals of semi-local Hecke indicators
AutomorphicForm.exists_forall_norm_sub_norm_mul_le_mul_rpow_mul_log_pow_of_isTwistedOrbitalIntegral_indicator_semiLocalIntegralSet_mul_mul34 below · cited by 2 · depth 34 - Uniform bound for twisted orbital integrals of a double coset
AutomorphicForm.exists_forall_norm_sub_norm_mul_le_of_isTwistedOrbitalIntegral_indicator_semiLocalIntegralSet_mul_mul35 below · cited by 1 · depth 34 - Uniform comparability of archimedean norms on a compact twisted window
AutomorphicForm.exists_forall_prod_norm_norm_pow_mult_le_mul_of_mem_of_isCompact5 below · cited by 1 · depth 34 - Torus pairing of matched Paley–Wiener profile with Eisenstein constant terms
AutomorphicForm.exists_forall_setIntegral_inv_ideleNorm_smul_integral_maximalCompact_mul_conj_constantTerm_eq_of_matched_paleyWiener11 below · cited by 1 · depth 34 - Maass–Selberg increment for two pairs of GL₂ Borel data
AutomorphicForm.exists_forall_setIntegral_lambdaT_pseudoEisenstein_mul_conj_sub_eq_maassSelberg_sub_and_sub_eq_twoTerm_sub_and_sub_eq_cross_sub_and_sub_eq_zero_two_pairs_slab129 below · cited by 1 · depth 34 - Eisenstein coefficients controlled by a weak L² distance
AutomorphicForm.exists_forall_tsum_integral_sum_normSq_setIntegral_axis_continuation_sub_le_mul_sq_of_forall_norm_setIntegral_sub_mul_conj_le1,003 below · cited by 1 · depth 34 - A compactly supported window for the local weighted discrepancy
AutomorphicForm.exists_hasCompactSupport_forall_norm_sub_le_forall_ratio_mul_sqrtRatio_mul_twistedWeighted_sub_finrank_mul_weighted_eq_of_areMatchingLocal85 below · cited by 2 · depth 34 - Local window functions for the finite places of S_K
AutomorphicForm.exists_hasCompactSupport_forall_norm_sub_one_mul_twistedWeighted_sub_finrank_mul_weighted_eq_mul_inv_ratio_mul_sqrtRatio_mul_of_areMatchingLocal87 below · cited by 1 · depth 34 - Archimedean covolume identity for a twisted commutant
AutomorphicForm.exists_isAddHaarMeasure_pi_infiniteBox_eq_one_and_sqrt_det_gram_mul_lintegral_archIdent_eq_sqrt_discr_pow_mul_norm_det_mul_lintegral_sum_map_tmul12 below · cited by 1 · depth 34 - Compact window for archimedean twisted orbital integrands
AutomorphicForm.exists_isCompact_forall_lintegral_lintegral_enorm_diagUnits2_unipotentGL2_sigmaGL_le_indicator5 below · cited by 1 · depth 34 - Properness of twisted conjugation modulo the twisted centralizer
AutomorphicForm.exists_isCompact_setOf_twistedConj_mem_subset_twistedCentralizer_mul9 below · cited by 1 · depth 34 - Descent of a semi-local twisted orbital integral to one place
AutomorphicForm.exists_isHaarMeasure_and_eq_integral_indicator_localIntegralSet_twistedConj_of_isTwistedOrbitalIntegral_indicator_semiLocalIntegralSet3 below · cited by 1 · depth 34 - Pseudo-Eisenstein approximation in the orthogonal complement of cusp forms
AutomorphicForm.exists_isSlabProfile_eLpNorm_sub_pseudoEisenstein_lt_of_forall_setIntegral_eq_zero_slab61 below · cited by 2 · depth 34 - Central translates of unit-factorisable test functions on GL₂(A_K)
AutomorphicForm.exists_isUnitFactorization_comp_centralScalar_mul0 below · cited by 1 · depth 34 - Euler factorisation of a weighted orbital integral on GL₂
AutomorphicForm.exists_isWeightedOrbitalIntegralOn_adeleRing_eq_mul_sum_prod_of_isUnitFactorization8 below · cited by 1 · depth 34 - Column map of a twisted commutant is an isomorphism after base change
AutomorphicForm.exists_linearEquiv_twistedCommutant_tensor_mulVec_tmul_of_mul_map_mem_center_of_forall_ne_scalar2 below · cited by 3 · depth 34 - Matched Paley–Wiener approximation of the non-cuspidal non-residual spectrum
AutomorphicForm.exists_matched_paleyWiener_forall_norm_setIntegral_sub_pseudoEisenstein_sub_mul_conj_le_of_orthogonal423 below · cited by 1 · depth 34 - Merging two matched Paley–Wiener packets with three-way decompositions
AutomorphicForm.exists_matched_paleyWiener_pair_eq_and_threeWay_of_matched_paleyWiener_of_matched_paleyWiener348 below · cited by 2 · depth 34 - Existence of a normalised Borel weight on GL₂(L⊗_K K_∞)
AutomorphicForm.exists_measurable_forall_integral_toTensorGL_diagUnits2_mul_diagUnits2_eq_one1 below · cited by 1 · depth 34 - Projection of an automorphic L² function onto a cut Hecke block
AutomorphicForm.exists_mem_isotypicCuspSubmodule_inf_archCutSubmodule_forall_convOp_eq_and_setIntegral_mul_conj_eq_of_forall_convOp_mem_principalLevel_of_isFundamentalDomain_slab351 below · cited by 1 · depth 34 - Local norm index at most two for a quadratic extension
AutomorphicForm.exists_mul_sigmaTensor_eq_includeRight_inv_mul_of_forall_ne_of_finrank_eq_two19 below · cited by 1 · depth 34 - Cellwise constant germ of the local weighted discrepancy
AutomorphicForm.exists_nhds_forall_eq_of_norm_sub_le_mul_norm_one_sub_forall_ratio_mul_sqrtRatio_mul_twistedWeighted_sub_finrank_mul_weighted_eq_of_areMatchingLocal77 below · cited by 3 · depth 34 - Local n-th roots near a scalar in a regular semisimple centraliser
AutomorphicForm.exists_nhds_forall_exists_pow_eq_of_isRegularSemisimple0 below · cited by 3 · depth 34 - Polynomial vertical bound for the continued intertwining operator
AutomorphicForm.exists_polynomial_bound_intertwining_continuation_of_isInducedSection159 below · cited by 2 · depth 34 - Archimedean twisted Harish-Chandra descent to the split torus
AutomorphicForm.exists_pos_forall_prod_norm_one_sub_norm_pow_mult_mul_lintegral_enorm_twistedConj_mul_eq_mul_lintegral_torus_unipotentGL2_rowIsometry22 below · cited by 1 · depth 34 - Weighted archimedean Harish–Chandra descent at the split torus
AutomorphicForm.exists_pos_forall_prod_norm_sub_one_pow_mul_eq_and_weighted_eq_mul_prod_norm_pow_mul_integral_integral_of_scalar_mul_diagUnits214 below · cited by 2 · depth 34 - Finiteness of the big-cell sum for a slab profile
AutomorphicForm.finite_support_pseudoEisenstein_summand10 below · cited by 11 · depth 34 - Idèle norm of det g splits over the places of K
AutomorphicForm.ideleNorm_det_map_genuineRingEquiv_eq_abs_algebraNorm_det_tensorArch_mul_prod_norm_algebraNorm_det_tensorPlace6 below · cited by 1 · depth 34 - Integrating out the centre over a determinant slab
AutomorphicForm.integrableOn_and_setIntegral_rationalTorusUnipotentQuotient_slab_mul_conj_eq_mul_setIntegral_inv_ideleNorm_smul_integral_maximalCompact9 below · cited by 1 · depth 34 - Twisted orbital integral of the unit at split diagonal δ
AutomorphicForm.integral_indicator_localIntegralSet_twistedConj_map_algEquiv_mul_eq_ite_inv_norm_sub_of_relIndex_eq5 below · cited by 1 · depth 34 - Twisted weighted orbital integral of the unit at an unramified place
AutomorphicForm.integral_indicator_localIntegralSet_twistedConj_map_algEquiv_mul_weight_eq_ite_finrank_mul_sum_of_relIndex_eq5 below · cited by 1 · depth 34 - Right invariance of the K-integral of modulus-equivariant functions
AutomorphicForm.integral_maximalCompact_comp_mul_eq_integral_of_forall_borel_mul_eq_modulus_mul9 below · cited by 3 · depth 34 - Unfolding the H-fibre of a twisted orbital integral
AutomorphicForm.integral_subgroup_centralScalar_twistedOrbital_mul_section_eq_const_mul_integral_ker_idelicNorm9 below · cited by 1 · depth 34 - Convolution by a level-N bi-invariant function is K_f-smooth
AutomorphicForm.isKfSmooth_convOp_and_apply_mul_eq_of_isBiInvariantUnder_principalLevel_of_ne_bot1 below · cited by 3 · depth 34 - Diagonal δ with distinct entry norms has regular semisimple norm string
AutomorphicForm.isRegularSemisimple_normString_of_diagonal_of_norm_ne3 below · cited by 1 · depth 34 - Central translation in twisted orbital integrals
AutomorphicForm.isTwistedOrbitalIntegralOn_comp_scalar_mul_iff0 below · cited by 1 · depth 34 - Diagonal twisted resolvent: units, cocycle relation, norm identity
AutomorphicForm.isUnit_and_mul_act_eq_add_and_prod_iterate_act_eq_norm_mul_of_mem_adelicBorel_of_diagonal0 below · cited by 2 · depth 34 - Axis limit of the convolved intertwining datum
AutomorphicForm.limUnder_nhdsNE_eq_convOp_axis_continuation_weylIntertwiningIntegral_of_meromorphicNFOn_of_eq_weylIntertwiningIntegral_convOp16 below · cited by 1 · depth 34 - Twisted commutant basis applied to a nonzero vector of L²
AutomorphicForm.linearIndependent_mulVec_and_span_eq_top_of_forall_isUnit_of_card_eq_four0 below · cited by 3 · depth 34 - Twisted-centralizer integral equals the local zeta factor
AutomorphicForm.lintegral_comp_conj_twistedCentralizer_eq_mul_inv_one_sub_mul_inv_one_sub_of_map_eq_smul_map_toTensorGL_localHaar2 below · cited by 2 · depth 34 - Determinant-shell integral over a twisted centralizer at a division place
AutomorphicForm.lintegral_iUnion_detShell_twistedCentralizer_eq_mul_inv_sub_one_mul_inv_one_sub_of_not_isSigmaConjugate_scalar_of_finrank_eq_two25 below · cited by 1 · depth 34 - Archimedean twisted orbital zeta integral at s=1
AutomorphicForm.lintegral_twistedCentralizer_mul_rpow_abs_algebraNorm_det_lt_top_and_tendsto_nhdsGT_one_of_map_coe_eq_smul_withDensity_gram0 below · cited by 1 · depth 34 - Haar volume and weighted integral of horocycle shells in GL₂(Kᵥ)
AutomorphicForm.localHaar_setOf_unipotentGL2_mul_eq_relIndex_and_setIntegral_weight_eq_of_norm_eq_inv0 below · cited by 4 · depth 34 - Adelic Haar pushforward along base change of a K-basis
AutomorphicForm.map_mulVec_sum_map_tmul_eq_measure_pi_adelicBox_smul_pi_of_linearIndependent_of_span_eq_top1 below · cited by 1 · depth 34 - Types of matrix coefficients of a right-translation-stable space
AutomorphicForm.matrixCoeff_mem_iSup_typeSubmodule_and_matrixCoeff_inv_mem_iSup_typeSubmodule_dual_of_forall_mem_iSup_typeSubmodule_comp0 below · cited by 2 · depth 34 - Twisted centraliser of a diagonal δ with norm ratio ≠ 1
AutomorphicForm.mem_sigmaCentralizer_iff_of_diagonal_of_norm_div_ne_one0 below · cited by 1 · depth 34 - Archimedean norm string of a global diagonal twisted class
AutomorphicForm.normString_tensorArch_eq_toTensorGL_diagUnits2_of_baseChangeGL_eq_globalPoints1 below · cited by 1 · depth 34 - Domination of twisted orbital integrals by weighted indicators
AutomorphicForm.norm_le_sum_mul_norm_of_isTwistedOrbitalIntegral_of_norm_le_sum_indicator6 below · cited by 1 · depth 34 - Right convolution transports a Paley–Wiener slab profile datum
AutomorphicForm.paleyWiener_convOp_and_convOp_pseudoEisenstein_eq_pseudoEisenstein_convOp_of_isArchBiFinite28 below · cited by 1 · depth 34 - Orthogonality extends to the L²-closure of the residual span
AutomorphicForm.setIntegral_mul_conj_eq_zero_of_forall_residualSpan_of_closure0 below · cited by 1 · depth 34 - Parseval identity for pseudo-Eisenstein series with residual term
AutomorphicForm.setIntegral_pseudoEisenstein_mul_conj_eq_inner_residualProj_add_sum_integral_axis_pairing_slab319 below · cited by 3 · depth 34 - Unfolding a pseudo-Eisenstein series against an automorphic function
AutomorphicForm.setIntegral_pseudoEisenstein_mul_conj_eq_setIntegral_rationalTorusUnipotentQuotient_slab43 below · cited by 5 · depth 34 - Unfolding a pseudo-Eisenstein series against a continuous automorphic function
AutomorphicForm.setIntegral_pseudoEisenstein_mul_conj_eq_setIntegral_rationalTorusUnipotentQuotient_slab_of_continuous52 below · cited by 1 · depth 34 - Local lattice covolume at a place of the first kind
AutomorphicForm.setLIntegral_lattice_norm_det_mul_norm_four_eq_mul_sqrt_norm_det_trace_of_map_conj_eq_smul_map_toTensorGL_localHaar25 below · cited by 1 · depth 34 - Local lattice covolume at a non-split place
AutomorphicForm.setLIntegral_lattice_norm_det_mul_norm_four_eq_mul_sqrt_norm_det_trace_of_not_isSigmaConjugate_scalar54 below · cited by 1 · depth 34 - Twisted orbital measure is additive Haar in lattice coordinates
AutomorphicForm.setLIntegral_twistedCentralizer_norm_det_mul_measure_pi_integers_eq_setLIntegral_lattice_mul_measure_preimage_of_isAddHaarMeasure10 below · cited by 3 · depth 34 - Unramified local zeta factor on a twisted centralizer
AutomorphicForm.setLIntegral_twistedCentralizer_semiLocalIntegers_eq_mul_inv_one_sub_absNorm_rpow_of_map_conj_eq_smul_map_toTensorGL_localHaar4 below · cited by 2 · depth 34 - Local twisted orbital integral over integral points equals (1-qᵥ⁻²)⁻¹(1-qᵥ⁻¹)⁻¹
AutomorphicForm.setLIntegral_twistedCentralizer_semiLocalIntegers_norm_det_eq_inv_one_sub_mul_inv_one_sub_of_map_conj_eq_map_toTensorGL_localHaar_of_measure_semiLocalIntegralSet_eq_one5 below · cited by 1 · depth 34 - Symmetric fold of the Eisenstein axis pairing
AutomorphicForm.sum_integral_axis_pairing_add_eq_half_mul_sum_integral_sum_conj_matrixCoeff_mul_fullCoeff_of_paleyWiener_matched362 below · cited by 1 · depth 34 - Right convolution commutes with the Weyl intertwining integral
AutomorphicForm.weylIntertwiningIntegral_convOp_eq_convOp_weylIntertwiningIntegral_of_isInducedSection_of_re_gt_half14 below · cited by 4 · depth 34 - Local lattice index at a division place of the twisted commutant
AutomorphicForm.absNorm_sq_mul_relIndex_sq_mul_norm_det_trace_eq_norm_sixteen_mul_relIndex_sq_of_forall_isUnit8 below · cited by 1 · depth 35 - Asymmetric axis pairing with intertwining term, matched data
AutomorphicForm.axis_pairing_add_inv_vol_axis_pairing_weylIntertwining_eq_sum_conj_matrixCoeff_mul_inner_mul_conj_of_paleyWiener_matched171 below · cited by 2 · depth 35 - Norm-conjugator carries M₂(Kᵥ) into the twisted commutant
AutomorphicForm.conj_map_includeRight_mem_twistedCommutant_of_map_conj_eq_smul_map_toTensorGL_localHaar0 below · cited by 1 · depth 35 - Right convolution preserves a Paley–Wiener family on GL₂
AutomorphicForm.continuous_and_differentiable_and_decay_and_eq_sum_integral_convOp_of_paleyWiener_family3 below · cited by 1 · depth 35 - Right convolution commutes with pseudo-Eisenstein series
AutomorphicForm.convOp_pseudoEisenstein_eq_pseudoEisenstein_convOp_of_isSlabProfile16 below · cited by 1 · depth 35 - Determinant of left multiplication on the local twisted commutant
AutomorphicForm.det_eq_algebraNorm_det_of_forall_mul_map_tmul_one_eq_sum_map_tmul_of_mem_twistedCommutant3 below · cited by 1 · depth 35 - L² bound for a residual projection on a truncation domain
AutomorphicForm.eLpNorm_residualProjection_le_eLpNorm_sub_of_forall_setIntegral_mul_conj_eq_zero0 below · cited by 1 · depth 35 - Merging two matched Paley–Wiener families over one index set
AutomorphicForm.exists_common_matched_paleyWiener_family_eq_sum_integral_of_matched_paleyWiener_of_matched_paleyWiener0 below · cited by 1 · depth 35 - Archimedean twisted weighted orbital germ expansion in (t,a)
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_archDisc_mul_twistedWeighted_eq_neg_two_mul_finrank_mul_sum_log_mul_twistedOrbital_add_sum_real_add_sum_complex75 below · cited by 1 · depth 35 - Archimedean weighted orbital germ identity at split classes
AutomorphicForm.exists_contDiff_hasCompactSupport_tsupport_subset_archDisc_mul_weighted_eq_neg_two_mul_sum_log_mul_orbital_add_sum_real_add_sum_complex35 below · cited by 1 · depth 35 - Level–type averaging kernel on the adelic maximal compact
AutomorphicForm.exists_continuous_idempotent_kernel_maximalCompact_comm_rowIsometry_levelTypeAverage_eq_self_and_mem_archCutSubmodule19 below · cited by 3 · depth 35 - Uniform finite-dimensional K_w-type for right convolutions of induced sections
AutomorphicForm.exists_finiteDimensional_forall_rightConv_mul_mem_of_isInducedSection_of_comp_inv_mem_archCutSubmodule3 below · cited by 1 · depth 35 - Second family of Eisenstein pairs as norm twists
AutomorphicForm.exists_forall_eq_mul_normPowChar_and_eq_mul_normPowChar_inv_of_pairs_of_exists_isInducedSection46 below · cited by 2 · depth 35 - Haar measure on GL₂(L⊗_K Kᵥ) in matrix coordinates
AutomorphicForm.exists_forall_lintegral_semiLocalHaar_eq_mul_lintegral_pi_norm_algebraNorm_det_inv_sq3 below · cited by 1 · depth 35 - Lower-unipotent absorption in GL₂(L⊗_K Kᵥ) with integrable weight
AutomorphicForm.exists_forall_lowerUnipotent_eq_diag_mul_unipotent_mul_mem_semiLocalIntegralSet6 below · cited by 1 · depth 35 - Bessel inequality for Eisenstein coefficients of an automorphised test function
AutomorphicForm.exists_forall_memLp_two_and_summable_and_tsum_integral_sum_normSq_setIntegral_finsum_integral_indicator_mul_conj_axis_continuation_le_mul_setIntegral_normSq981 below · cited by 2 · depth 35 - Vanishing of automorphised compactly supported functions above a height
AutomorphicForm.exists_forall_mem_canonicalTruncationDomain_finsum_integral_indicator_eq_zero_of_lt_adelicHeight23 below · cited by 2 · depth 35 - High-height vanishing of the pseudo-Eisenstein series of a slab profile
AutomorphicForm.exists_forall_mem_canonicalTruncationDomain_pseudoEisenstein_eq_zero_of_lt_adelicHeight0 below · cited by 3 · depth 35 - A single function computes twisted weighted orbital integrals off t=1
AutomorphicForm.exists_forall_nhds_eq_isCompact_forall_isTwistedWeightedOrbitalIntegral_diagUnits2_eq_of_isSemiLocalTestFn21 below · cited by 1 · depth 35 - Uniform shell bound for upper-triangular slices of Hecke double cosets
AutomorphicForm.exists_forall_setLIntegral_withDensity_norm_inv_iSup_measure_setOf_upperTriangular_mem_doubleCoset_le8 below · cited by 1 · depth 35 - Polarised Paley–Wiener identity for axis pairings against a matched packet
AutomorphicForm.exists_forall_tsum_integral_sum_axisPairing_mul_conj_axisPairing_pseudoEisenstein_eq_mul_setIntegral_mul_conj_sub_residualProj_of_matched_paleyWiener_of_lt_adelicHeight966 below · cited by 1 · depth 35 - Uniform fibre-volume bound for σ-twisted conjugation on L⊗_K Kᵥ
AutomorphicForm.exists_forall_withDensity_norm_inv_setOf_norm_mem_and_mul_sigmaTensor_eq_mul_mul_le13 below · cited by 1 · depth 35 - Descent of twisted orbital integrals along the twisted shift
AutomorphicForm.exists_integral_indicator_pi_twistedShift_mul_eq_integral_indicator_mul_of_forall_integral_eq_one2 below · cited by 1 · depth 35 - Residual projection on the canonical truncation domain
AutomorphicForm.exists_isAutomorphicFnAt_residualProjection_of_isAutomorphicFnAt_canonicalTruncationDomain24 below · cited by 3 · depth 35 - Properness of a ↦ σ(a)a⁻¹ modulo K_∞^×
AutomorphicForm.exists_isCompact_forall_exists_includeRight_mul_mem_of_sigmaTensor_mul_inv_mem0 below · cited by 1 · depth 35 - Non-zero swapped induced section on the unitary axis
AutomorphicForm.exists_isInducedSection_swap_ne_zero_of_isInducedSection_family_principalLevel_archCutSubmodule_of_apply_ne_zero336 below · cited by 2 · depth 35 - Regularity of the continued intertwining operator on Re s≥ 0
AutomorphicForm.exists_isOpen_analyticOnNhd_continuousOn_intertwining_continuation_of_isInducedSection137 below · cited by 1 · depth 35 - Archimedean matching at a regular split diagonal element
AutomorphicForm.exists_isOrbitalIntegralOn_and_exists_isTwistedOrbitalIntegralOn_and_eq_of_areMatchingArch_diagUnits26 below · cited by 1 · depth 35 - Density of Paley–Wiener slab profiles on a determinant slab
AutomorphicForm.exists_isSlabProfile_paleyWiener_eLpNorm_sub_restrict_rationalTorusUnipotentQuotient_lt_of_isSlabProfile117 below · cited by 2 · depth 35 - Determinant of x ↦ Aσ(x)-Bx on L⊗_K Kᵥ
AutomorphicForm.exists_linearMap_mul_sigmaTensor_sub_mul_and_det_eq_neg_one_pow_mul_norm_sub_norm3 below · cited by 1 · depth 35 - Iwasawa integration formula on GL₂(L ⊗_K K_∞)
AutomorphicForm.exists_lintegral_tensor_infiniteAdeleRing_eq_mul_lintegral_diagUnits2_unipotentGL2_archIdentGL_rowIsometry13 below · cited by 2 · depth 35 - Matched Paley–Wiener approximation of Eisenstein coefficients of an automorphisation
AutomorphicForm.exists_matched_paleyWiener_tsum_integral_sum_normSq_setIntegral_mul_conj_axis_continuation_sub_le994 below · cited by 1 · depth 35 - Bounded Borel section for the diagonal torus in GL₂
AutomorphicForm.exists_measurable_forall_integral_localCentralizer_toTensorGL_mul_eq_one_of_diagonal0 below · cited by 1 · depth 35 - Twisted weighted orbital germ near t=1 at a finite place
AutomorphicForm.exists_nhds_forall_eq_of_norm_sub_le_and_norm_add_halfWeighted_sub_le_and_forall_ratio_mul_sqrtRatio_mul_twistedWeighted_eq_of_areMatchingLocal73 below · cited by 1 · depth 35 - Regularised polynomial vertical bound for the continued intertwining operator
AutomorphicForm.exists_polynomial_bound_regularized_intertwining_continuation_of_isInducedSection158 below · cited by 1 · depth 35 - Three-way slab decomposition of a pseudo-Eisenstein series
AutomorphicForm.exists_threeWay_principalLevel_archCutSubmodule_ae_eq_pseudoEisenstein_sub_residualProjection_slab117 below · cited by 1 · depth 35 - Adelic twisted centralizer element with prescribed components on S
AutomorphicForm.exists_twistedCentralizer_coe_eq_sum_map_tmul_and_tensorPlace_eq_one_of_forall_exists8 below · cited by 1 · depth 35 - Torus Whittaker coefficients of a Bruhat–Eisenstein series via factorisation datum
AutomorphicForm.exists_whittakerCoefficient_bruhatEisenstein_diagOne_eq_cpowChar_mul_sum_eulerProduct_of_factorizationDatum_one58 below · cited by 1 · depth 35 - Windowed section for the diagonal torus acting on GL₂(L⊗_K Kᵥ)
AutomorphicForm.exists_windowed_section_localCentralizer_toTensorGL_of_diagonal0 below · cited by 1 · depth 35 - Torus pairing of a matched Paley–Wiener packet by Mellin inversion
AutomorphicForm.integrable_and_setIntegral_inv_ideleNorm_smul_integral_lineIntegral_mul_conj_eq_of_isInducedSection_of_eq_mul_normPowChar5 below · cited by 1 · depth 35 - Vanishing torus pairing for separated induced sections
AutomorphicForm.integrable_and_setIntegral_inv_ideleNorm_smul_integral_lineIntegral_mul_conj_eq_zero_of_isInducedSection_of_apply_ne7 below · cited by 1 · depth 35 - Integrability of the axis pairings of a matched Paley–Wiener datum
AutomorphicForm.integrable_axis_pairing_convOp_add_inv_vol_axis_pairing_convOp_weylIntertwining_of_paleyWiener_matched16 below · cited by 1 · depth 35 - Integrability of θ_{|φ|}· f on a slab fundamental domain
AutomorphicForm.integrable_pseudoEisenstein_norm_mul_restrict_of_isLsXiFunction_of_continuous34 below · cited by 1 · depth 35 - Twisted conjugate of a diagonal element in Iwasawa coordinates
AutomorphicForm.inv_mul_diagUnits2_mul_sigmaGL_of_diagUnits2_mul_unipotentGL2_mul0 below · cited by 5 · depth 35 - Right convolution with an archimedean-type test function is K-finite
AutomorphicForm.isArchKFinite_rightConv_of_isInducedSection_of_comp_inv_mem_archCutSubmodule3 below · cited by 4 · depth 35 - The torus chart (z,a)↦ zcdotbc(diag(a,1)) is a closed embedding
AutomorphicForm.isClosedEmbedding_centralScalar_mul_baseChangeGL_toTensorGL_diagUnits21 below · cited by 1 · depth 35 - Non-zero elements of the local twisted commutant are units
AutomorphicForm.isUnit_of_mem_twistedCommutant_map_of_ne_zero_of_not_isSigmaConjugate_scalar_tensorPlace3 below · cited by 2 · depth 35 - Level–type averaging fixes L² automorphic vectors almost everywhere
AutomorphicForm.levelTypeAverage_ae_eq_self_of_isAutomorphicFnAt_of_mem_archCutSubmodule46 below · cited by 3 · depth 35 - Archimedean twisted commutant as a real span of xᵢ⊗ωₐ
AutomorphicForm.linearIndependent_and_coe_span_map_tmul_integralBasis_eq_setOf_mul_eq_mul_map_sigmaTensor1 below · cited by 2 · depth 35 - Twisted orbital integrand at a diagonal element in big-cell coordinates
AutomorphicForm.lintegral_enorm_twistedConj_mul_semiLocalHaar_eq_mul_lintegral_lintegral_torus_unipotentChart_of_isTwistedSectionFnOn_of_diagonal6 below · cited by 1 · depth 35 - Section-weight swap in archimedean Iwasawa coordinates
AutomorphicForm.lintegral_mul_eq_mul_lintegral_torus_unipotentGL2_rowIsometry_of_forall_lintegral_toTensorGL_diagUnits2_mul_eq_one5 below · cited by 1 · depth 35 - Haar-nullity of the norm-zero locus in M₂(L⊗_K Kᵥ)
AutomorphicForm.measure_setOf_algebraNorm_det_sum_map_tmul_eq_zero_eq_zero_of_isUnit1 below · cited by 3 · depth 35 - Twisted centralizer at w contains almost all coordinate vectors
AutomorphicForm.measure_setOf_not_exists_twistedCentralizer_coe_eq_sum_map_tmul_eq_zero2 below · cited by 1 · depth 35 - Valuation shells of the twisted centraliser all have equal measure
AutomorphicForm.measure_setOf_valuation_det_eq_exp_neg_twistedCentralizer_eq_measure_setOf_valuation_det_eq_one_of_isNormOf_scalar_of_finrank_eq_two22 below · cited by 1 · depth 35 - Square-summability of the Eisenstein coefficients of a matched Paley–Wiener profile
AutomorphicForm.memLp_two_and_summable_integral_sum_normSq_setIntegral_pseudoEisenstein_mul_conj_axis_continuation_of_matched_paleyWiener290 below · cited by 1 · depth 35 - Semi-local GL₂ above a finite place splits over w∣ v
AutomorphicForm.mem_semiLocalIntegralSet_iff_and_semiLocalHaar_doubleCoset_localEmbed_eq_localHaar_and_map_baseChangeAlgEquiv_eq_smul_pi0 below · cited by 9 · depth 35 - Archimedean log-height and its Weyl translate
AutomorphicForm.neg_log_archHeight_archIdentGL_sub_log_archHeight_adelicWeyl_mul_eq_sum_mult_mul_log0 below · cited by 3 · depth 35 - Entry bounds on a double coset KrhoK in GL₂(Kᵥ)
AutomorphicForm.norm_apply_sq_le_localHaar_doubleCoset_mul_norm_det1 below · cited by 1 · depth 35 - Transport of Paley–Wiener data by a maximal-compact kernel average
AutomorphicForm.paleyWiener_levelTypeAverage_and_pseudoEisenstein_levelTypeAverage_eq_and_residualProjection_of_kernel_maximalCompact_detOne47 below · cited by 2 · depth 35 - Archimedean module equals absolute real algebra norm
AutomorphicForm.prod_norm_archIdent_pow_mult_eq_abs_algebraNorm_real2 below · cited by 1 · depth 35 - Archimedean Jacobian of ξ↦σ(ξ)-λ'ξ
AutomorphicForm.prod_norm_one_sub_norm_pow_mult_mul_lintegral_comp_sigmaTensor_sub_mul_eq_lintegral5 below · cited by 1 · depth 35 - Level and archimedean type of a pseudo-Eisenstein series
AutomorphicForm.pseudoEisenstein_principalLevel_and_mem_archCutSubmodule_of_paleyWiener_principalLevel_archCutSubmodule12 below · cited by 2 · depth 35 - Local index identity between a lattice and a conjugated matrix order
AutomorphicForm.relIndex_conj_map_integers_sq_mul_norm_det_trace_eq_norm_sixteen_mul_relIndex_sq5 below · cited by 1 · depth 35 - Idempotent compact average is self-adjoint and contractive
AutomorphicForm.setIntegral_levelTypeAverage_mul_conj_eq_and_eLpNorm_levelTypeAverage_le_of_isAutomorphicFnAt_of_idempotent_kernel_maximalCompact29 below · cited by 3 · depth 35 - Parseval identity for pseudo-Eisenstein series on a determinant slab
AutomorphicForm.setIntegral_pseudoEisenstein_mul_conj_eq_sum_integral_maximalCompact_pairing_slab52 below · cited by 1 · depth 35 - Residual pairing for pseudo-Eisenstein series over a determinant slab
AutomorphicForm.setIntegral_residualProj_mul_conj_eq_sum_integral_maximalCompact_residue_pairing_slab261 below · cited by 1 · depth 35 - Index scaling between two lattices in the twisted commutant
AutomorphicForm.setLIntegral_lattice_norm_det_mul_relIndex_eq_setLIntegral_closure_conj_mul_relIndex13 below · cited by 1 · depth 35 - Index scaling between lattice and integral-determinant order
AutomorphicForm.setLIntegral_lattice_norm_det_mul_relIndex_eq_setLIntegral_closure_det_mem_integers_mul_relIndex_of_not_isSigmaConjugate_scalar20 below · cited by 1 · depth 35 - Mass of a conjugated maximal order at a finite place
AutomorphicForm.setLIntegral_mem_closure_conj_map_integers_norm_det_eq_mul_inv_one_sub_mul_inv_one_sub4 below · cited by 1 · depth 35 - Local twisted orbital mass over the integral-determinant order
AutomorphicForm.setLIntegral_mem_closure_det_mem_integers_norm_det_eq_mul_inv_sub_one_mul_inv_one_sub_of_not_isSigmaConjugate_scalar29 below · cited by 1 · depth 35 - Unramified local zeta integral of M₂(mathcal Oᵥ)
AutomorphicForm.setLIntegral_nnnorm_det_rpow_setOf_integral_eq_measure_localIntegralSet_mul1 below · cited by 2 · depth 35 - Archimedean covolume of the twisted commutant lattice
AutomorphicForm.sqrt_det_gram_smul_map_volume_image_parallelepiped_tmul_integralBasis_eq_sqrt_discr_pow_mul_norm_det9 below · cited by 1 · depth 35 - Involution identity for the matched Paley–Wiener fold
AutomorphicForm.sum_conj_matrixCoeff_mul_axis_pairing_weylIntertwining_mul_conj_fullCoeff_eq_axis_pairing_swap_neg_of_paleyWiener_matched360 below · cited by 1 · depth 35 - Twisted centraliser of a diagonal δ with distinct norms
AutomorphicForm.twistedCentralizer_eq_map_toTensorGL_localCentralizer_and_exists_isHaarMeasure_map_eq_of_diagonal_of_norm_ne3 below · cited by 1 · depth 35 - Compact kernel averages: a.e. additivity and an L² bound
AutomorphicForm.ae_levelTypeAverage_sub_eq_and_eLpNorm_levelTypeAverage_le_of_isAutomorphicFnAt_of_kernel_maximalCompact30 below · cited by 1 · depth 36 - Functional equation of the Eisenstein axis continuation, frame form
AutomorphicForm.axis_continuation_eq_sum_inner_weylIntertwining_mul_axis_continuation_of_swap_normPowChar613 below · cited by 3 · depth 36 - Smoothness and unit-carried support of |N_∞|^s-twists
AutomorphicForm.contDiff_and_hasCompactSupport_prod_norm_archEval_pow_rpow_mul_of_tsupport0 below · cited by 2 · depth 36 - Continuity and polynomial growth of Eisenstein coefficients of θ_Ψ
AutomorphicForm.continuous_setIntegral_finsum_integral_indicator_mul_conj_axis_continuation_and_exists_norm_le_mul_one_add_abs_pow418 below · cited by 1 · depth 36 - R(f) on the axis continuation of the Weyl intertwining integral
AutomorphicForm.convOp_axis_continuation_weylIntertwiningIntegral_eq_sum_integral_rightConv_mul_conj_mul_axis_continuation_weylIntertwiningIntegral94 below · cited by 1 · depth 36 - R(f) acts on continued intertwining integrals by matrix coefficients
AutomorphicForm.convOp_axis_continuation_weylIntertwiningIntegral_eq_sum_mul_axis_continuation_weylIntertwiningIntegral_light94 below · cited by 1 · depth 36 - Gram determinant of the archimedean trace form on M_m(L⊗_K K_∞)
AutomorphicForm.det_trace_real_matrix_trace_map_tmul_mul_eq_discr_pow_mul_norm_det4 below · cited by 1 · depth 36 - Integral reduced norms give a rank-four lattice of discriminant ‖16‖qᵥ⁻²
AutomorphicForm.exists_closure_iff_det_mem_integers_and_norm_det_trace_mul_absNorm_sq_eq_norm_sixteen_of_forall_isUnit6 below · cited by 1 · depth 36 - Smooth compactly supported descent of archimedean orbital averages
AutomorphicForm.exists_contDiff_hasCompactSupport_forall_integral_conj_diagUnits2_mul_unipotentGL2_eq_of_isCompact7 below · cited by 1 · depth 36 - Smoothness of the twisted K-average of an archimedean test factor
AutomorphicForm.exists_contDiff_hasCompactSupport_forall_integral_twistedConj_diagUnits2_mul_unipotentGL2_eq7 below · cited by 1 · depth 36 - Mixed logarithmic potential at a real place
AutomorphicForm.exists_contDiff_hasCompactSupport_integral_mul_log_norm_one_sub_sq_add_norm_sq_eq_add_norm_mul_of_isReal13 below · cited by 1 · depth 36 - Archimedean logarithmic potential at a complex place
AutomorphicForm.exists_contDiff_hasCompactSupport_integral_mul_log_norm_one_sub_sq_add_norm_sq_eq_add_norm_sq_mul_log_mul_of_isComplex11 below · cited by 1 · depth 36 - Twisted log-weight layer above a real place of K
AutomorphicForm.exists_contDiff_hasCompactSupport_prod_norm_pow_mul_integral_ker_norm_integral_twistedLogWeight_eq_add_norm_mul_of_isReal33 below · cited by 1 · depth 36 - Twisted log-weight layer above a complex place of K
AutomorphicForm.exists_contDiff_hasCompactSupport_prod_norm_pow_mul_integral_ker_norm_integral_twistedLogWeight_eq_add_norm_sq_mul_log_mul_of_isComplex25 below · cited by 1 · depth 36 - Straightening a θ-twisted cyclic shift on G^{m+1}
AutomorphicForm.exists_continuousMulEquiv_sigmaCentralizer_homeomorph_measurePreserving_twistedShift1 below · cited by 1 · depth 36 - Self-adjoint convolution unit for a finite-dimensional translation-stable subspace
AutomorphicForm.exists_continuous_convolution_idempotent_forall_integral_mul_apply_eq_of_finiteDimensional_of_star_mem0 below · cited by 2 · depth 36 - Finite-dimensional level–type orbit space on the maximal compact
AutomorphicForm.exists_finiteDimensional_biInvariant_levelTypeOrbitSubmodule_maximalCompact_detOne8 below · cited by 2 · depth 36 - Finitely many Kᵥ^× G₀-cosets of twist-bounded units
AutomorphicForm.exists_finset_forall_eq_mul_algebraMap_mul_of_sigmaTensor_eq_mul5 below · cited by 2 · depth 36 - Span transport between normalised axis continuation and swapped block
AutomorphicForm.exists_forall_inv_vol_mul_axis_continuation_weylIntertwining_eq_sum_and_exists_forall_eq_sum_of_paleyWiener_matched_swap275 below · cited by 2 · depth 36 - Slab L² norm bounded by L² norm in Iwasawa coordinates
AutomorphicForm.exists_forall_isSlabProfile_eLpNorm_sub_restrict_rationalTorusUnipotentQuotient_le_mul_eLpNorm_sub_diagOne_mul9 below · cited by 1 · depth 36 - Uniform germ bound for twisted weighted orbital integrals at t=1
AutomorphicForm.exists_forall_norm_ratio_mul_sqrtRatio_mul_twistedWeighted_add_halfWeighted_sub_le_of_areMatchingLocal62 below · cited by 1 · depth 36 - Wave-packet form of a matched Paley–Wiener pseudo-Eisenstein series
AutomorphicForm.exists_forall_pseudoEisenstein_sub_residualProj_ae_eq_mul_sum_integral_sum_inner_mul_axis_continuation_of_matched_paleyWiener957 below · cited by 2 · depth 36 - L²-density of continuous K_∞-finite automorphic functions
AutomorphicForm.exists_isAutomorphicFnAt_continuous_isArchKFinite_principalLevel_archCutSubmodule_eLpNorm_sub_lt_of_isAutomorphicFnAt45 below · cited by 1 · depth 36 - Relative compactness of the height-truncated truncation domain
AutomorphicForm.exists_isCompact_canonicalTruncationDomain_inter_setOf_adelicHeight_le_subset22 below · cited by 5 · depth 36 - A compact open subgroup of norm-one units in L⊗_K Kᵥ
AutomorphicForm.exists_isCompact_isOpen_one_mem_mul_mem_norm_eq_one_tensor_adicCompletion1 below · cited by 2 · depth 36 - Uniform Haar normalisation on split regular centralisers over K_∞
AutomorphicForm.exists_isHaarMeasure_forall_integral_centralizer_diagUnits2_eq_integral_prod_of_map_eq0 below · cited by 2 · depth 36 - Normalised Haar measure on a twisted centraliser, with orbital integral
AutomorphicForm.exists_isHaarMeasure_twistedCentralizer_and_exists_isTwistedWeightedOrbitalIntegral_of_normString_diagUnits2_eq4 below · cited by 1 · depth 36 - Induced sections replace K-finite factors without increasing L² distance
AutomorphicForm.exists_isInducedSection_eLpNorm_sub_sum_mul_restrict_maximalCompact_le_of_isSlabProfile31 below · cited by 1 · depth 36 - Regularity across Re s=0 of the L-normalised GL(2) intertwining operator
AutomorphicForm.exists_isOpen_analyticOnNhd_continuousOn_eulerProduct_mul_intertwining_continuation121 below · cited by 2 · depth 36 - L² approximation of slab profiles by Paley–Wiener profiles
AutomorphicForm.exists_isSlabProfile_paleyWiener_eLpNorm_sub_lt_of_forall_eLpNorm_le_of_dense0 below · cited by 1 · depth 36 - Determinant of y↦ aσ(y)-by on L⊗_K A
AutomorphicForm.exists_linearMap_apply_eq_mul_sigmaTensor_sub_mul_and_det_eq_neg_one_pow_mul_norm_sub_norm_of_infinite1 below · cited by 2 · depth 36 - Matched Paley–Wiener data realising prescribed smooth coefficient families
AutomorphicForm.exists_matched_paleyWiener_sum_integral_sum_conj_inner_mul_eq_sum_integral_sum_conj_integral_mul_cexp_mul_and_setIntegral_normSq_sub_residualProj_le_of_contDiff_hasCompactSupport465 below · cited by 1 · depth 36 - Symmetric square-summable families approximated by Paley–Wiener coefficients
AutomorphicForm.exists_matched_paleyWiener_tsum_integral_sum_normSq_sub_setIntegral_axis_continuation_le_of_symmetric484 below · cited by 1 · depth 36 - A common principal level for a holomorphic K_f-smooth family
AutomorphicForm.exists_ne_bot_forall_apply_mul_eq_of_mem_principalLevel_of_isKfSmooth_of_differentiable2 below · cited by 3 · depth 36 - Uniform cells for twisted lifts and normalised weighted orbital values
AutomorphicForm.exists_nhds_forall_iff_and_ratio_mul_sqrtRatio_mul_twistedWeighted_eq_of_norm_sub_le_of_areMatchingLocal22 below · cited by 1 · depth 36 - Small diagonal shifts fix the σ-twisted integrand
AutomorphicForm.exists_nhds_nhds_forall_apply_inv_mul_diagUnits2_mul_toTensorGL_diagUnits2_mul_sigmaGL_eq_of_isSemiLocalTestFn6 below · cited by 1 · depth 36 - One-index Paley–Wiener slab profile with prescribed torus values
AutomorphicForm.exists_paleyWiener_oneIndex_apply_diagOne_mul_eq_of_isInducedSection_of_contDiff_of_pos13 below · cited by 1 · depth 36 - Swap-closed separated normal form for summed Paley–Wiener data
AutomorphicForm.exists_paleyWiener_swapClosed_separated_eq_sum_of_forall_paleyWiener48 below · cited by 1 · depth 36 - Polynomial vertical growth of the L-regularised intertwining operator
AutomorphicForm.exists_polynomial_bound_eulerProduct_mul_intertwining_continuation_of_isInducedSection145 below · cited by 1 · depth 36 - Archimedean unfolding of twisted and weighted twisted orbital integrals
AutomorphicForm.exists_pos_forall_twistedOrbital_archHaarL_diagUnits2_eq_mul_integral_ker_norm_integral_integral_and_twistedWeighted_eq_of_coupled33 below · cited by 1 · depth 36 - Residual projection of a level-N type vector: existence and a.e. uniqueness
AutomorphicForm.exists_residualProjection_mem_span_chiDet_principalLevel_archCutSubmodule_and_ae_eq_of_isAutomorphicFnAt76 below · cited by 2 · depth 36 - L² approximation of a slab profile by elementary tensors
AutomorphicForm.exists_sum_character_mul_smooth_mul_kFinite_eLpNorm_sub_lt_of_isSlabProfile32 below · cited by 1 · depth 36 - Semi-local Cartan type and volume bound for KaK
AutomorphicForm.exists_uniformizers_forall_exists_cartanType_mem_doubleCoset_and_prod_pow_le_semiLocalHaar4 below · cited by 1 · depth 36 - Polynomial local L² bounds for unitary GL₂ Eisenstein series
AutomorphicForm.forall_exists_eLpNorm_axis_continuation_restrict_le_mul_pow_of_isCompact_of_ne_bot413 below · cited by 1 · depth 36 - Locally uniform moderate growth of continued Eisenstein series
AutomorphicForm.forall_isCompact_exists_forall_norm_axis_continuation_le_mul_adelicHeight_rpow_of_mem_canonicalTruncationDomain132 below · cited by 2 · depth 36 - Adjoint transport of the axis intertwining operator
AutomorphicForm.integral_mul_conj_eq_integral_axis_continuation_weylIntertwining_mul_conj_axis_continuation_weylIntertwining_of_paleyWiener_matched334 below · cited by 1 · depth 36 - Matrix expansion of the right-convolution pairing over K
AutomorphicForm.integral_mul_conj_rightConv_eq_sum_conj_inner_mul_inner_of_orthonormal_span_of_isInducedSection_of_isArchBiFinite28 below · cited by 1 · depth 36 - Right K-translates and kernel averages of automorphic L² members
AutomorphicForm.isAutomorphicFnAt_comp_mul_and_eLpNorm_eq_and_eLpNorm_levelTypeAverage_le_of_kernel_maximalCompact28 below · cited by 3 · depth 36 - Norm string equal to the base matrix implies norm
AutomorphicForm.isNormOf_of_normString_eq_toTensorGL0 below · cited by 1 · depth 36 - Norm-string identity at a finite place: scalar is a σ-norm
AutomorphicForm.isNormOf_scalar_tensorPlace_of_normString_eq_toTensorGL_centralScalar0 below · cited by 2 · depth 36 - Compact kernel average preserves slab profiles and Paley–Wiener form
AutomorphicForm.isSlabProfile_levelTypeAverage_and_eq_sum_integral_of_kernel_maximalCompact5 below · cited by 1 · depth 36 - Pseudo-Eisenstein series commute with right compact averages
AutomorphicForm.levelTypeAverage_pseudoEisenstein_eq_pseudoEisenstein_levelTypeAverage_of_isSlabProfile16 below · cited by 1 · depth 36 - Haar measure of GL₂(L⊗_K Kᵥ) in big-cell coordinates
AutomorphicForm.lintegral_semiLocalHaar_eq_mul_lintegral_lintegral_torus_mul_unipotentChart4 below · cited by 1 · depth 36 - Archimedean module of y ↦ aσ(y) - by on L ⊗_K K_∞
AutomorphicForm.map_mul_sigmaTensor_sub_mul_addHaar_infiniteAdeleRing_eq_inv_prod_norm_archEval_algebraNorm_sub_pow_mult_smul4 below · cited by 2 · depth 36 - Volume of integral matrices of determinant valuation k
AutomorphicForm.measure_setOf_integral_valuation_det_eq_geom_sum_absNorm_mul_measure_localIntegralSet0 below · cited by 1 · depth 36 - Norm on L⊗_K Kᵥ is the product of local norms
AutomorphicForm.norm_algebraNorm_eq_prod_norm_baseChangeAlgEquiv_apply1 below · cited by 1 · depth 36 - Orthonormality and swapped section law of normalised intertwined family
AutomorphicForm.orthonormal_and_isInducedSection_inv_vol_mul_axis_continuation_weylIntertwiningIntegral_of_flat_orthonormal_family274 below · cited by 10 · depth 36 - Compact kernel averaging preserves Paley–Wiener families of induced sections
AutomorphicForm.paleyWiener_sections_levelTypeAverage_of_kernel_maximalCompact_detOne7 below · cited by 1 · depth 36 - Archimedean modulus of a K_∞-linear map on L⊗_K K_∞
AutomorphicForm.prod_norm_pow_mult_mul_lintegral_comp_linearMap_tensor_infiniteAdeleRing_eq_lintegral_of_det_eq2 below · cited by 1 · depth 36 - Lift-independence of twisted weighted orbital integrals at diag(a,at)
AutomorphicForm.ratio_mul_sqrtRatio_mul_twistedWeighted_eq_of_normString_diagUnits2_eq_of_areMatchingLocal5 below · cited by 1 · depth 36 - Residue at s=1/2 of the self-dual Weyl intertwining continuation
AutomorphicForm.residue_weylIntertwining_continuation_self_dual_eq_div_measure_slab_mul_maximalCompact_pairing_mul_det259 below · cited by 1 · depth 36 - Symmetry of Eisenstein coefficients under the axis functional equation
AutomorphicForm.setIntegral_mul_conj_axis_continuation_eq_sum_conj_inner_weylIntertwining_mul_setIntegral_of_swap_normPowChar618 below · cited by 1 · depth 36 - Term-by-term pairing of truncated automorphic function with Eisenstein packet
AutomorphicForm.setIntegral_mul_conj_sum_integral_sum_inner_mul_axis_continuation_eq_sum_integral_sum_conj_inner_mul_setIntegral_of_isAutomorphicFnAt_of_lt_adelicHeight416 below · cited by 3 · depth 36 - Pseudo-Eisenstein inner product over a determinant slab
AutomorphicForm.setIntegral_pseudoEisenstein_mul_conj_pseudoEisenstein_eq_setIntegral_quotient_slab47 below · cited by 1 · depth 36 - Weak functional equation for Eisenstein coefficients of a height-capped vector
AutomorphicForm.sum_integral_sum_conj_inner_weylIntertwining_mul_setIntegral_mul_conj_axis_continuation_eq_of_isAutomorphicFnAt_of_lt_adelicHeight733 below · cited by 1 · depth 36 - Unipotent fibre of a GL₂ Cartan double coset and its measure bound
AutomorphicForm.unipotentGL2_conj_diagonal_mem_doubleCoset_iff_and_norm_sub_mul_measureReal_le1 below · cited by 1 · depth 36 - Unit-invariance and compact finiteness of the norm-weighted Haar measure
AutomorphicForm.withDensity_norm_inv_preimage_mul_eq_and_lt_top_of_isCompact3 below · cited by 1 · depth 36 - Integral twisted commutant closed under ring operations
AutomorphicForm.zero_mem_and_one_mem_and_add_mem_and_neg_mem_and_mul_mem_and_smul_mem_maximalOrder_twistedCommutant_of_not_isSigmaConjugate_scalar2 below · cited by 2 · depth 36 - Almost-everywhere uniqueness of residual projections on truncation domains
AutomorphicForm.ae_eq_of_residualProjection_of_residualProjection_canonicalTruncationDomain0 below · cited by 1 · depth 37 - Pullback of type submodules along a compatible homomorphism
AutomorphicForm.comp_mem_iSup_typeSubmodule_of_mem_iSup_typeSubmodule_of_comp_eq0 below · cited by 1 · depth 37 - Constant term of a pseudo-Eisenstein series of a slab profile
AutomorphicForm.constantTerm_pseudoEisenstein_eq_add_weylIntertwiningIntegral14 below · cited by 1 · depth 37 - Continuity of a Paley–Wiener slab profile and its pseudo-Eisenstein series
AutomorphicForm.continuous_and_continuous_pseudoEisenstein_of_paleyWiener_slabProfile16 below · cited by 1 · depth 37 - Regularity of the unitary-axis Eisenstein wave packet
AutomorphicForm.continuous_and_isLsXiFunction_and_isKfSmooth_and_principalLevel_and_mem_archCutSubmodule_sum_integral_sum_inner_mul_axis_continuation_of_matched_paleyWiener430 below · cited by 1 · depth 37 - Continuity and polynomial growth of Eisenstein coefficients
AutomorphicForm.continuous_setIntegral_mul_conj_axis_continuation_and_exists_norm_le_mul_one_add_abs_pow_of_isAutomorphicFnAt_of_lt_adelicHeight416 below · cited by 1 · depth 37 - Right convolution preserves level, type cut and K_∞-finiteness
AutomorphicForm.convOp_principalLevel_invariant_and_mem_archCutSubmodule_and_isArchKFinite_of_isAutomorphicFnAt40 below · cited by 1 · depth 37 - Approximate identity bound for R(f)v-v on a truncation domain
AutomorphicForm.eLpNorm_convOp_sub_le_of_forall_eLpNorm_comp_mul_sub_le_of_isAutomorphicFnAt26 below · cited by 1 · depth 37 - Twisting by ‖·‖^{iτ} shifts the η-parameter
AutomorphicForm.etaFst_etaSnd_mul_normPowChar_eq_shift0 below · cited by 4 · depth 37 - Normalised Weyl intertwining integral on K: continuation past Re s=0
AutomorphicForm.exists_analyticOnNhd_continuousOn_normalisedIntertwining_of_isInducedSection_family37 below · cited by 1 · depth 37 - Central idele correction making all archimedean determinants one
AutomorphicForm.exists_centralScalar_mem_adelicMaximalCompact_det_archComponent_mul_eq_one0 below · cited by 1 · depth 37 - Twisted archimedean log integral at a complex place over a real place
AutomorphicForm.exists_contDiff_hasCompactSupport_integral_ker_norm_integral_mul_log_sq_add_norm_resolvent_sq_eq_add_norm_mul_of_isComplex_place23 below · cited by 1 · depth 37 - Twisted log integral at a real place: A+lVert 1-trVert_w B
AutomorphicForm.exists_contDiff_hasCompactSupport_integral_ker_norm_integral_mul_log_sq_add_norm_resolvent_sq_eq_add_norm_mul_of_isReal_place18 below · cited by 1 · depth 37 - Complex-over-complex archimedean log layer for twisted resolvents
AutomorphicForm.exists_contDiff_hasCompactSupport_integral_ker_norm_integral_mul_log_sq_add_norm_resolvent_sq_eq_add_norm_sq_mul_log_mul_of_isComplex_isComplex17 below · cited by 1 · depth 37 - Approximate identity at principal level on GL₂(A_K)
AutomorphicForm.exists_continuous_hasCompactSupport_integral_eq_one_principalLevel_conj_invariant_subset_nhds0 below · cited by 1 · depth 37 - L² approximation of a slab profile by continuous band-supported functions
AutomorphicForm.exists_continuous_invariant_bandSupported_eLpNorm_sub_lt_of_isSlabProfile27 below · cited by 1 · depth 37 - Iwasawa factorisation of Haar measure on GL₂(L⊗_K K_∞)
AutomorphicForm.exists_eq_smul_map_diagUnits2_mul_unipotentGL2_mul_and_integral_eq_of_isHaarMeasure_tensor_infiniteAdeleRing14 below · cited by 1 · depth 37 - Equivariant K-finite averaging for slab profiles
AutomorphicForm.exists_equivariant_kFinite_eLpNorm_sub_sum_mul_le_of_isSlabProfile26 below · cited by 1 · depth 37 - Finitely many principal-level cosets cover the finite maximal compact
AutomorphicForm.exists_finset_adelicMaximalCompact_finiteAdelic_coset_principalLevel0 below · cited by 1 · depth 37 - Vanishing germ of the half-weighted orbital integral at non-norm parameters
AutomorphicForm.exists_forall_norm_halfWeighted_sub_le_of_not_exists_norm_eq_of_areMatchingLocal10 below · cited by 1 · depth 37 - Twisted minus untwisted weighted orbital germ at t=1
AutomorphicForm.exists_forall_norm_ratio_mul_sqrtRatio_mul_twistedWeighted_add_halfWeighted_sub_le_of_normString_diagUnits2_eq_of_areMatchingLocal56 below · cited by 1 · depth 37 - Weak wave-packet identity against pseudo-Eisenstein test series
AutomorphicForm.exists_forall_setIntegral_pseudoEisenstein_mul_conj_sub_residualProj_sub_mul_sum_integral_sum_inner_mul_axis_continuation_eq_zero_of_matched_paleyWiener_of_cuspBasis563 below · cited by 1 · depth 37 - Iwasawa extension of an equivariant K-finite function to an induced section
AutomorphicForm.exists_isInducedSection_continuous_forall_maximalCompact_eq_of_equivariant_kFinite4 below · cited by 1 · depth 37 - Splitting the archimedean twisted log-weighted integral
AutomorphicForm.exists_linearMap_prod_norm_pow_mul_integral_comp_sigmaTensor_sub_mul_twistedLogWeight_eq_add_sum9 below · cited by 2 · depth 37 - Matched Paley–Wiener data realising prescribed smooth coefficient families
AutomorphicForm.exists_matched_paleyWiener_injective_and_inner_eq_integral_mul_cexp_and_sum_integral_sum_conj_inner_mul_eq_and_setIntegral_normSq_sub_residualProj_le_of_contDiff_hasCompactSupport464 below · cited by 2 · depth 37 - Principal congruence levels are cofinal in neighbourhoods of 1
AutomorphicForm.exists_nat_principalLevel_inf_finiteAdelicGL2Subgroup_subset_of_mem_nhds_one1 below · cited by 2 · depth 37 - Local constancy of diagonal norm-string lifts near t=1
AutomorphicForm.exists_nhds_forall_exists_normString_diagUnits2_eq_toTensorGL_iff_of_norm_sub_le5 below · cited by 1 · depth 37 - Near t=1, normalised twisted weighted values agree on cells
AutomorphicForm.exists_nhds_forall_ratio_mul_sqrtRatio_mul_twistedWeighted_eq_of_norm_sub_le_of_normString_diagUnits2_eq_of_areMatchingLocal16 below · cited by 1 · depth 37 - Strong L²-continuity of right translation at the identity
AutomorphicForm.exists_nhds_one_forall_eLpNorm_comp_mul_sub_lt_of_isAutomorphicFnAt_canonicalTruncationDomain31 below · cited by 1 · depth 37 - Diagonal norm strings over Kᵥ realise exactly pairs of local norms
AutomorphicForm.exists_normString_diagUnits2_eq_toTensorGL_diagUnits2_iff_exists_norm_eq3 below · cited by 3 · depth 37 - Twisted commutant division algebra: uniformiser norm and unit trace
AutomorphicForm.exists_norm_det_mul_absNorm_eq_one_and_exists_norm_trace_eq_one_of_forall_isUnit5 below · cited by 1 · depth 37 - Strip bound for intertwining operator times Hecke Euler product
AutomorphicForm.exists_norm_eulerProduct_mul_intertwining_le_mul_one_add_norm_eulerProduct_of_isInducedSection141 below · cited by 1 · depth 37 - Archimedean twisted fibration over the norm-one torus
AutomorphicForm.exists_pos_forall_lintegral_units_tensor_eq_mul_lintegral_ker_norm_of_forall_lintegral_mul_includeRight_eq13 below · cited by 1 · depth 37 - Residual projection onto the level-N good lines
AutomorphicForm.exists_residualProjection_mem_span_chiDet_principalLevel_of_isAutomorphicFnAt33 below · cited by 1 · depth 37 - L² approximation of band-supported invariant functions by elementary tensors
AutomorphicForm.exists_sum_character_mul_smooth_mul_kFinite_eLpNorm_sub_lt_of_continuous_invariant_bandSupported20 below · cited by 1 · depth 37 - Trace and determinant on a twisted commutant are Kᵥ-scalars
AutomorphicForm.exists_trace_eq_one_tmul_and_det_eq_one_tmul_and_norm_sq_le_of_forall_isUnit0 below · cited by 2 · depth 37 - Hilbert 90 for L⊗_K Kᵥ over a cyclic extension
AutomorphicForm.exists_units_mul_sigmaTensor_eq_of_norm_eq_one4 below · cited by 3 · depth 37 - Finite-dimensionality of the multi-place isotypic intersection
AutomorphicForm.finiteDimensional_iInf_iSup_typeSubmodule_mulSingle0 below · cited by 1 · depth 37 - Vanishing of axis Eisenstein combinations with zero constant term
AutomorphicForm.forall_axis_continuation_sub_sum_mul_axis_continuation_eq_zero_of_forall_constantTerm_eq_zero533 below · cited by 1 · depth 37 - Uniform L² bound for truncated Eisenstein series on the unitary axis
AutomorphicForm.forall_exists_setIntegral_norm_sq_lambdaT_axis_continuation_le_mul_mul_pow_of_isArchCompAt_of_ne_bot409 below · cited by 2 · depth 37 - Orthogonality extends from Paley–Wiener data to all slab profiles
AutomorphicForm.forall_isSlabProfile_setIntegral_pseudoEisenstein_mul_conj_eq_zero_of_forall_matched_paleyWiener_setIntegral_pseudoEisenstein_mul_conj_eq_zero418 below · cited by 1 · depth 37 - Arbitrary-family pseudo-Eisenstein orthogonality from sum-extension orthogonality
AutomorphicForm.forall_matched_paleyWiener_setIntegral_pseudoEisenstein_mul_conj_eq_zero_of_forall_sum_extension_setIntegral_pseudoEisenstein_mul_conj_eq_zero1 below · cited by 1 · depth 37 - Parseval expansion of the intertwined Weyl coefficient over K
AutomorphicForm.inner_weylIntertwining_eq_sum_inner_mul_conj_inner_of_matched_paleyWiener0 below · cited by 1 · depth 37 - Integrability of the twisted archimedean descent integrand
AutomorphicForm.integrable_archIdentGL_inv_mul_diagUnits2_mul_unipotentGL2_sigmaTensor_sub_mul_sigmaGL_mul_of_isArchTestFactor_of_isRegularSemisimple8 below · cited by 1 · depth 37 - Adjointness of right convolution for the maximal compact pairing
AutomorphicForm.integral_maximalCompactHaar_rightConv_mul_conj_eq_integral_mul_conj_rightConv_star_of_isInducedSection_axis12 below · cited by 1 · depth 37 - Inversion of normalised intertwining operators on the unitary axis
AutomorphicForm.inv_vol_sum_inner_axis_continuation_weylIntertwiningIntegral_mul_eq_self_of_swap_normPowChar332 below · cited by 1 · depth 37 - Uniform finite-dimensional translate spaces give archimedean K-finiteness
AutomorphicForm.isArchKFinite_of_forall_exists_finiteDimensional_forall_mem0 below · cited by 3 · depth 37 - Factorizable test functions are stable under f↦̄f(·⁻¹)
AutomorphicForm.isFactorizableTestFn_conj_comp_inv6 below · cited by 1 · depth 37 - Right convolution preserves induced sections with level and type
AutomorphicForm.isInducedSection_rightConv_and_continuous_and_isArchKFinite_and_principalLevel_and_mem_archCutSubmodule_of_isArchBiFinite7 below · cited by 1 · depth 37 - From almost-everywhere to pointwise cuspidality and K_f-smoothness
AutomorphicForm.isSmoothCuspAutomorphicFnAt_of_continuous_of_principalLevel_of_ae_constantTerm_eq_zero3 below · cited by 1 · depth 37 - Square-integrability of the axis Eisenstein wave packet
AutomorphicForm.memLp_two_restrict_canonicalTruncationDomain_sum_integral_sum_inner_mul_axis_continuation_of_matched_paleyWiener478 below · cited by 1 · depth 37 - Good-line combination orthogonal to the residual span lies in the cut
AutomorphicForm.mem_archCutSubmodule_of_mem_span_chiDet_principalLevel_of_residualProjection70 below · cited by 1 · depth 37 - Convolution of a flat section expands in the orthonormal family
AutomorphicForm.rightConv_eq_sum_integral_rightConv_mul_conj_mul_of_orthonormal_complete_flat_family13 below · cited by 1 · depth 37 - Expansion of R(f)φ_{e,j,s} in a flat orthonormal family
AutomorphicForm.rightConv_eq_sum_integral_rightConv_mul_conj_mul_of_orthonormal_complete_flat_family_light13 below · cited by 1 · depth 37 - Orthogonality of matched wave packets to the cusp basis
AutomorphicForm.setIntegral_sum_integral_sum_inner_mul_axis_continuation_mul_conj_cuspBasis_eq_zero_of_matched_paleyWiener713 below · cited by 1 · depth 37 - Conjugate-inverse involution preserves the archimedean type cut
AutomorphicForm.star_mem_archCutSubmodule_of_finiteDimensional_of_forall_comp_mul_mem_of_support1 below · cited by 1 · depth 37 - Bessel bound for K-pairings of intertwined and flat families
AutomorphicForm.sum_norm_sq_sum_conj_inner_weylIntertwining_mul_le_sum_norm_sq_of_matched_paleyWiener277 below · cited by 1 · depth 37 - The canonical truncation domain has positive Haar measure
AutomorphicForm.adelicGLHaar_canonicalTruncationDomain_pos21 below · cited by 1 · depth 38 - Archimedean type cut is preserved by the continued Eisenstein series
AutomorphicForm.axis_continuation_bruhatEisenstein_mem_archCutSubmodule_of_forall_mem_archCutSubmodule2 below · cited by 2 · depth 38 - Right U(N)-invariance of the continued Eisenstein series
AutomorphicForm.axis_continuation_bruhatEisenstein_mul_principalLevel_eq_of_isArchKFinite_family0 below · cited by 4 · depth 38 - Continuity, automorphy and level of a matched Eisenstein wave packet
AutomorphicForm.continuous_and_isLsXiFunction_and_principalLevel_sum_integral_sum_inner_mul_axis_continuation_of_matched_paleyWiener415 below · cited by 1 · depth 38 - A common matched Paley–Wiener family carrying two section families
AutomorphicForm.exists_common_matched_paleyWiener_family_eq_sum_integral_and_sections_eq_of_matched_paleyWiener_of_matched_paleyWiener_light0 below · cited by 1 · depth 38 - Norm-fibre integral of an archimedean test function is smooth
AutomorphicForm.exists_contDiff_hasCompactSupport_forall_apply_norm_eq_integral_ker_norm_prod_of_contDiff2 below · cited by 3 · depth 38 - Coordinate split at a real place for twisted resolvents
AutomorphicForm.exists_continuousLinearEquiv_forall_norm_archEval_resolvent_eq_abs_fst_add_of_isReal5 below · cited by 1 · depth 38 - Split complex place: coordinates linearising the archimedean resolvent
AutomorphicForm.exists_continuousLinearEquiv_forall_norm_archEval_resolvent_eq_norm_fst_add_of_isComplex_of_isComplex_comap5 below · cited by 1 · depth 38 - Twisted resolvent at a complex place over a real place
AutomorphicForm.exists_continuousLinearEquiv_norm_archEval_resolvent_eq_norm_conj_add_conj_mul_of_isComplex_of_isReal_comap7 below · cited by 1 · depth 38 - Continuous F^×-invariant functions are L²-dense
AutomorphicForm.exists_continuous_forall_principalIdeles_eLpNorm_sub_lt_of_memLp_withDensity_ideleNorm_inv_prod_maximalCompactHaar11 below · cited by 1 · depth 38 - Level–type averaging kernel on the adelic maximal compact
AutomorphicForm.exists_continuous_idempotent_kernel_maximalCompact_comm_rowIsometry_levelTypeAverage_eq_self_and_mem_archCutSubmodule_of_continuous19 below · cited by 1 · depth 38 - Square-integrability of the truncated Eisenstein constant-term packet
AutomorphicForm.exists_forall_memLp_two_indicator_highSet_sum_integral_sum_inner_mul_add_inv_vol_mul_axis_continuation_weylIntertwining_of_matched_paleyWiener339 below · cited by 1 · depth 38 - L²-boundedness of the truncated Eisenstein wave packet
AutomorphicForm.exists_forall_memLp_two_lambdaT_sum_integral_sum_inner_mul_axis_continuation_restrict_canonicalTruncationDomain_of_matched_paleyWiener451 below · cited by 1 · depth 38 - Logarithmic expansion of twisted weighted orbital values
AutomorphicForm.exists_forall_norm_ratio_mul_sqrtRatio_mul_twistedWeighted_add_mul_log_mul_twistedOrbital_sub_le_of_normString_diagUnits2_eq44 below · cited by 1 · depth 38 - Plancherel identity for the continuous part in canonical Eisenstein coordinates
AutomorphicForm.exists_forall_setIntegral_normSq_pseudoEisenstein_sub_residualProj_eq_mul_sum_integral_sum_normSq_inner_axis_of_matched_paleyWiener449 below · cited by 1 · depth 38 - Maass–Selberg L² bound for truncated GL₂ Eisenstein series
AutomorphicForm.exists_forall_setIntegral_norm_sq_lambdaT_axis_continuation_le_mul_pow_of_eq_or_exists_normOneIdeles_of_isArchCompAt_of_ne_bot408 below · cited by 1 · depth 38 - Twisted resolvent and change of variables for y↦σ y-ry
AutomorphicForm.exists_linearMap_resolvent_integral_comp_sigmaTensor_sub_mul_eq_integral_mul_comp_smul7 below · cited by 1 · depth 38 - Growth of the Weyl intertwining operator at large height
AutomorphicForm.exists_norm_eulerProduct_mul_intertwining_le_mul_norm_eulerProduct_of_le_abs_im_of_isInducedSection35 below · cited by 1 · depth 38 - Bounded strips: relative bound for the continued intertwining operator
AutomorphicForm.exists_norm_eulerProduct_mul_intertwining_le_mul_of_abs_im_le_of_isInducedSection139 below · cited by 1 · depth 38 - Swap partners with norm-power twist for complete Eisenstein families
AutomorphicForm.exists_partner_forall_partner_partner_eq_and_eq_mul_normPowChar_of_pairs_complete275 below · cited by 1 · depth 38 - Uniform approximation of band-supported invariant functions by elementary tensors
AutomorphicForm.exists_sum_character_mul_smooth_mul_kFinite_norm_sub_le_of_continuous_invariant_bandSupported7 below · cited by 1 · depth 38 - Hilbert 90 for norm-one units of L ⊗_K K_∞
AutomorphicForm.exists_units_eq_inv_mul_unitsMap_sigmaTensor_of_norm_eq_one3 below · cited by 1 · depth 38 - Rapid decay of isotypic cusp forms on a determinant slab
AutomorphicForm.forall_exists_forall_norm_le_mul_inv_adelicHeight_pow_of_mem_isotypicCuspSubmodule_principal_inf_archCutSubmodule345 below · cited by 4 · depth 38 - Right-invariance of the K-pairing for axis induced sections
AutomorphicForm.integral_maximalCompactHaar_mul_apply_mul_conj_eq_of_isInducedSection_axis_of_isUnitaryChar10 below · cited by 1 · depth 38 - Averaging kernel of level and type fixes the Eisenstein packet
AutomorphicForm.integral_maximalCompact_mul_sum_integral_sum_inner_mul_axis_continuation_eq_self_of_levelTypeAverage_eq_self417 below · cited by 1 · depth 38 - The character line χ∘det is ξ-automorphic on the truncation domain
AutomorphicForm.isAutomorphicFnAt_chiDet_of_squaresToXi_of_continuous20 below · cited by 1 · depth 38 - K_∞^× → (L⊗_K K_∞)^× is a closed embedding
AutomorphicForm.isClosedEmbedding_unitsMap_includeRight0 below · cited by 1 · depth 38 - Entire combinations and imaginary shifts of flat induced-section families
AutomorphicForm.isInducedSection_and_continuous_and_isArchKFinite_and_isKfSmooth_sum_mul_shift_of_flat_family1 below · cited by 1 · depth 38 - K_f-smoothness of the continued Eisenstein series
AutomorphicForm.isKfSmooth_axis_continuation_bruhatEisenstein_of_principalLevel_of_isArchKFinite_family2 below · cited by 1 · depth 38 - Mellin packets of flat induced sections are slab profiles
AutomorphicForm.isSlabProfile_and_forall_eq_sum_integral_of_paleyWiener_packet14 below · cited by 1 · depth 38 - Unweighted twisted orbital identity at a diagonal lift
AutomorphicForm.ratio_mul_eq_splitOrbital_of_isTwistedOrbitalIntegral_of_normString_diagUnits2_eq_of_areMatchingLocal11 below · cited by 1 · depth 38 - Continued Eisenstein series pair to zero with the cuspidal basis
AutomorphicForm.setIntegral_axis_continuation_mul_conj_cuspBasis_eq_zero_of_mem524 below · cited by 2 · depth 38 - Explicit two-sided quasi-inverse of σ - r on L⊗_K A
AutomorphicForm.sigmaTensor_twistedResolvent_sub_mul_eq_one_sub_norm_smul_and_twistedResolvent_sigmaTensor_sub_mul_eq1 below · cited by 5 · depth 38 - Truncation of an Eisenstein wave packet on the unitary axis
AutomorphicForm.sum_integral_sum_inner_mul_axis_continuation_sub_lambdaT_eq_indicator_highSet_sum_integral_sum_inner_mul_add_inv_vol_mul_weylIntertwining_of_matched_paleyWiener414 below · cited by 1 · depth 38 - Local log splitting above an archimedean place of K
AutomorphicForm.sum_mult_mul_log_one_add_norm_archEval_archIdent_smul_inv_eq_and_sum_mult_eq_finrank_mul0 below · cited by 2 · depth 38 - σ⊗ 1-invariant units of L⊗_K K_∞ are 1⊗ K_∞^×
AutomorphicForm.unitsMap_sigmaTensor_eq_self_iff_mem_range_unitsMap_includeRight0 below · cited by 1 · depth 38 - Holomorphy of the Eisenstein–cusp pairing on the truncation domain
AutomorphicForm.analyticOnNhd_setIntegral_axis_continuation_mul_conj_cuspBasis426 below · cited by 1 · depth 39 - Transport of places under g⊗ 1 on L⊗_K K_∞
AutomorphicForm.archEval_archIdent_sigmaTensor_eq_mapRingHom_and_ringEquiv_mixedSpace_fst_eq_and_snd_eq_or_eq_conj1 below · cited by 3 · depth 39 - Continuity, integrability and L²-ness of wave-packet coefficients
AutomorphicForm.continuous_and_integrable_and_memLp_two_integral_mul_conj_flat_section_of_matched_paleyWiener0 below · cited by 3 · depth 39 - Uniform L²(K) bounds for the normalised intertwining operator
AutomorphicForm.exists_forall_integral_maximalCompact_norm_sq_deriv_axis_continuation_weylIntertwiningIntegral_le_and_norm_sq_sub_le_of_flat_of_ne_bot404 below · cited by 1 · depth 39 - Flat induced-section families are bounded on vertical strips
AutomorphicForm.exists_forall_norm_apply_le_of_isInducedSection_etaFst_etaSnd_of_flat_of_isUnitaryChar2 below · cited by 1 · depth 39 - Twisted weight asymptotics for the family σ-u
AutomorphicForm.exists_forall_norm_mul_integral_comp_sigmaTensor_sub_smul_mul_semiLocalWeight_add_sub_integral_mul_log_norm_trace_le12 below · cited by 1 · depth 39 - Uniform L² bound for truncated unitary Eisenstein series
AutomorphicForm.exists_forall_setIntegral_norm_sq_lambdaT_axis_continuation_le_of_flat_of_eLpNorm_deriv_le272 below · cited by 1 · depth 39 - Iwasawa majorant bound on the truncated high-height region
AutomorphicForm.exists_forall_setLIntegral_canonicalTruncationDomain_inter_lt_adelicHeight_le_mul_lintegral_Ioi_of_le_ideleNorm_mul44 below · cited by 1 · depth 39 - Iwasawa unfolding of J'+cI' at a diagonal twisted element
AutomorphicForm.exists_pos_forall_integrable_and_twistedWeighted_add_mul_twistedOrbital_eq_mul_integral_iwasawa_of_normString_diagUnits2_eq15 below · cited by 1 · depth 39 - A uniform measurable torus section at a finite place
AutomorphicForm.exists_torusSection_forall_normString_diagUnits2_eq18 below · cited by 1 · depth 39 - Iwasawa law for flat sections and continued intertwining amplitudes
AutomorphicForm.flat_section_centralScalar_mul_diagOne_mul_eq_mul_ideleNorm_cpow_and_inv_vol_mul_axis_continuation_weylIntertwining_eq_and_rationalTorusUnipotent_mul_of_matched_paleyWiener275 below · cited by 1 · depth 39 - Vertical-line integral of a flat induced-section packet
AutomorphicForm.integrable_and_integral_mellin_mul_apply_borel_mul_eq_of_isInducedSection_of_flat2 below · cited by 1 · depth 39 - Pointwise Parseval on the unitary axis for induced sections
AutomorphicForm.integral_normSq_add_inv_vol_limUnder_weylIntertwining_eq_sum_normSq_inner_axis_of_orthonormal_span_principalLevel222 below · cited by 1 · depth 39 - Truncation commutes with the Eisenstein wave-packet integral
AutomorphicForm.lambdaT_sum_integral_sum_inner_mul_axis_continuation_eq_sum_integral_sum_inner_mul_lambdaT_of_mem_canonicalTruncationDomain_of_matched_paleyWiener414 below · cited by 1 · depth 39 - Mellin L²-finiteness of the two constant-term amplitudes
AutomorphicForm.lintegral_maximalCompact_lintegral_Ioi_enorm_sq_integral_sum_inner_mul_cpow_mul_flat_section_add_inv_vol_mul_axis_continuation_weylIntertwining_mul_inv_lt_top_of_matched_paleyWiener331 below · cited by 1 · depth 39 - Eisenstein continuation orthogonal to cusp basis for Re s>1/2
AutomorphicForm.setIntegral_axis_continuation_mul_conj_cuspBasis_eq_zero_of_re_gt_half520 below · cited by 1 · depth 39 - Plancherel identity for the continuous part of a pseudo-Eisenstein series
AutomorphicForm.setIntegral_normSq_pseudoEisenstein_sub_residualProj_eq_mul_sum_integral_normSq_add_weylIntertwining_of_principalLevel_slab440 below · cited by 1 · depth 39 - Off-axis K-expansion of matched Paley–Wiener sections
AutomorphicForm.differentiable_inner_and_decay_and_eq_sum_inner_mul_flat_orthonormal_of_matched_paleyWiener0 below · cited by 1 · depth 40 - Iwasawa-coordinate integration formula for GL₂(L⊗_K Kᵥ)
AutomorphicForm.exists_forall_lintegral_semiLocalHaar_eq_mul_lintegral_torus_unipotentGL2_setLIntegral_semiLocalIntegralSet4 below · cited by 2 · depth 40 - Moderate growth of the absolute Bruhat series on the truncation domain
AutomorphicForm.exists_forall_norm_add_tsum_norm_le_mul_adelicHeight_rpow_of_isInducedSection_of_mem_canonicalTruncationDomain242 below · cited by 1 · depth 40 - High-cusp unfolding bound in Iwasawa coordinates
AutomorphicForm.exists_forall_setLIntegral_canonicalTruncationDomain_inter_lt_adelicHeight_le_mul_setLIntegral_iwasawa_indicator33 below · cited by 1 · depth 40 - Properness of a↦σ(a)a⁻¹ above a finite place
AutomorphicForm.exists_isCompact_forall_exists_includeRight_mul_mem_of_sigmaTensor_mul_inv_mem_adicCompletion8 below · cited by 1 · depth 40 - Iwasawa coordinates for Haar measure on GL₂(L⊗_K Kᵥ)
AutomorphicForm.exists_pos_forall_integral_semiLocalHaar_eq_mul_integral_integral_setIntegral_iwasawa5 below · cited by 1 · depth 40 - Axis pairings of a Paley–Wiener family: integrability and reflection
AutomorphicForm.integrable_and_eq_axis_pairings_normalForm_weylIntertwining_of_paleyWiener_family332 below · cited by 1 · depth 40 - Torus sections give equal ‖N‖⁻¹-weighted integrals
AutomorphicForm.lintegral_norm_inv_mul_torusSection_mul_eq_of_forall_mul_eq7 below · cited by 1 · depth 40 - Haar scaling of y ↦ σ(y) - ry on L ⊗_K Kᵥ
AutomorphicForm.map_sigmaTensor_sub_mul_eq_inv_nnnorm_one_sub_norm_smul5 below · cited by 1 · depth 40 - Left-diagonal and right-integral invariance of the semi-local weight
AutomorphicForm.semiLocalWeight_diagUnits2_mul_mul_eq_of_mem_semiLocalIntegralSet0 below · cited by 1 · depth 40 - Semi-local weight of a unipotent at a finite place
AutomorphicForm.semiLocalWeight_unipotentGL2_eq_finsum_log_max_and_eq_log_norm_norm_of_one_le_and_norm_baseChangeAlgEquiv_algebraMap_eq_pow1 below · cited by 1 · depth 40 - Module of x ↦ a σ(x) - bx on L ⊗_K Kᵥ
AutomorphicForm.map_mul_sigmaTensor_sub_mul_eq_norm_algebraNorm_sub_inv_smul4 below · cited by 1 · depth 41
AutomorphicForm.AdelicTracePushforward 3
- Haar compatibility of the local trace push-forward
AutomorphicForm.AdelicTracePushforward.exists_pos_forall_integral_localTracePushforward_eq_mul_integral0 below · cited by 1 · depth 28 - Local trace push-forward is adjoint to composition with the trace
AutomorphicForm.AdelicTracePushforward.exists_pos_forall_integral_localTracePushforward_mul_eq_mul_integral_mul_comp_trace0 below · cited by 1 · depth 28 - Trace of trace-adapted adelic coordinates equals r
AutomorphicForm.AdelicTracePushforward.trace_traceFibre0 below · cited by 1 · depth 33
AutomorphicForm.ArchOccursInClassOf 1
- Conjugation-equivariance of archimedean occurrence in an eigensystem class
AutomorphicForm.ArchOccursInClassOf.map_starRingEnd0 below · cited by 1 · depth 20
AutomorphicForm.ArchWeightOne 2
- Weight-one normalised holomorphy forces central exponent ω(t)=t
AutomorphicForm.ArchWeightOne.central_eq_coe_of_forall_mdifferentiable0 below · cited by 1 · depth 17 - Weight-one Fourier coefficient of a right translate
AutomorphicForm.ArchWeightOne.intervalIntegral_exp_neg_mul_translate_rotation_eq0 below · cited by 1 · depth 17
AutomorphicForm.ClassSumGrowth 2
- Growth of class sums of Hecke recursion values
AutomorphicForm.ClassSumGrowth.exists_forall_classBlock_le_and_classSum_le135 below · cited by 2 · depth 18 - Linear lower bound for class-restricted mean-square Hecke sums
AutomorphicForm.ClassSumGrowth.exists_forall_le_classSum_of_classCarriesMass487 below · cited by 1 · depth 18
AutomorphicForm.ComplexIwasawa 12
- Smoothness and bounded derivatives of the Iwasawa compact factor
AutomorphicForm.ComplexIwasawa.contDiff_and_exists_bound_iteratedFDeriv_kC_apply0 below · cited by 3 · depth 23 - Continuity, holomorphy and rapid decay of j_{a,b}
AutomorphicForm.ComplexIwasawa.continuousOn_differentiableOn_norm_le_polyDecay_weightFourierIntegral5 below · cited by 7 · depth 23 - The compact factor kC g z lies in SU(2)
AutomorphicForm.ComplexIwasawa.kC_mem_specialUnitaryGroup0 below · cited by 2 · depth 23 - Polynomial decay of the Fourier transform of (radC g)^{-u}P
AutomorphicForm.ComplexIwasawa.norm_fourierIntegral_cpow_radC_mul_le_polyDecay0 below · cited by 1 · depth 23 - Explicit Iwasawa factorisation of w n(z) g in GL₂(ℂ)
AutomorphicForm.ComplexIwasawa.weyl_mul_unipotent_mul_eq_borel_mul_kC0 below · cited by 1 · depth 23 - Uniform rapid decay of Fourier integrals of radC^{-u}P
AutomorphicForm.ComplexIwasawa.exists_forall_norm_fourierIntegral_cpow_radC_mul_le_polyDecay_of_isCompact4 below · cited by 1 · depth 24 - Smoothness and uniform symbol bounds for rad_ℂ^{-u}
AutomorphicForm.ComplexIwasawa.contDiff_and_exists_forall_bound_iteratedFDeriv_cpow_neg_radC_of_isCompact2 below · cited by 1 · depth 25 - Uniform bound for area integrals of rad_ℂ^{-κ}
AutomorphicForm.ComplexIwasawa.exists_forall_integrable_integral_rpow_neg_radC_le_of_isCompact0 below · cited by 1 · depth 25 - Uniform derivative bounds for the compact Iwasawa factor on compacta
AutomorphicForm.ComplexIwasawa.exists_forall_bound_iteratedFDeriv_kC_apply_of_isCompact0 below · cited by 1 · depth 26 - Entire continuation and decay of the complex-place Jacquet integral
AutomorphicForm.ComplexIwasawa.exists_entire_weightFourierIntegral_norm_le_rpow_neg_mul_polyDecay6 below · cited by 1 · depth 30 - Entire continuation of the complex-place weight–Fourier integral
AutomorphicForm.ComplexIwasawa.exists_entire_eq_weightFourierIntegral_norm_le_of_ne_zero6 below · cited by 2 · depth 32 - Uniform strip bound for the complex-place weight–Fourier integral
AutomorphicForm.ComplexIwasawa.exists_forall_norm_weightFourierIntegral_continuation_le_mul_pow_abs_im_mul_pow_of_re_mem_Icc6 below · cited by 1 · depth 34
AutomorphicForm.CuspidalConstituent 103
- Local Whittaker space at p: irreducible, admissible, smooth
AutomorphicForm.CuspidalConstituent.IsCuspConstituent.localSpaceAt_cyclic_finite_fixed_smooth_of_hasMultiplicityOneAt182 below · cited by 3 · depth 17 - Twisting a cuspidal constituent by a finite-order Hecke character
AutomorphicForm.CuspidalConstituent.exists_cuspConstituentMeets_span_image_fnTwist_of_isIsotypicCuspFormAt_of_isBoundedGenuineFn_of_forall_not_dvd23 below · cited by 1 · depth 17 - Finite-adelic translates of one type-χ vector span the others
AutomorphicForm.CuspidalConstituent.mem_span_rightTranslate_finiteAdelic_of_isCuspConstituent_of_finiteDimensional_of_mem_levelInvariantSubmodule_of_mem_archCutSubmodule_ofChar_of_pos82 below · cited by 1 · depth 17 - Local component at a finite place of a cuspidal constituent
AutomorphicForm.CuspidalConstituent.IsCuspConstituent.exists_irreducible_admissible_isotypicAt171 below · cited by 4 · depth 18 - J-stability of a cuspidal constituent at a real place
AutomorphicForm.CuspidalConstituent.comp_mul_archRealGLAt_J_mem_of_isCuspConstituent_of_cuspConstituentMeets_of_coversModCentre241 below · cited by 6 · depth 18 - Members of the K_∞-finite cuspidal span are continuous cusp forms
AutomorphicForm.CuspidalConstituent.continuous_and_isSmoothCuspAutomorphicFnAt_rightTranslate_of_mem_cuspKFiniteSubmodule0 below · cited by 19 · depth 18 - Cuspidal constituents sharing a nonzero vector coincide
AutomorphicForm.CuspidalConstituent.eq_of_isCuspConstituent_of_exists_mem_ne_zero0 below · cited by 2 · depth 18 - Finite eigenexpansion of isotypic cusp vectors under one test function
AutomorphicForm.CuspidalConstituent.exists_eq_sum_rightConv_eq_smul_of_mem_isotypicCuspSubmodule_inf_archCutSubmodule329 below · cited by 2 · depth 18 - Scalar action of conjugation-invariant test functions on a one-type cut
AutomorphicForm.CuspidalConstituent.exists_forall_rightConv_eq_smul_of_isCuspConstituent_of_finiteDimensional_ofChar_of_pos74 below · cited by 1 · depth 18 - Finite eigencapture of a level-and-type cut of a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_inf_levelInvariantSubmodule_inf_archCutSubmodule_le_iSup_rightConv_eq_smul_of_isCuspConstituent87 below · cited by 1 · depth 18 - Cut vectors of a cuspidal constituent are bounded by ‖det‖^{w₀/2}
AutomorphicForm.CuspidalConstituent.exists_norm_le_mul_ideleNorm_det_rpow_of_isCuspConstituent173 below · cited by 5 · depth 18 - Level-and-type cut distributes over finite sums of cusp subrepresentations
AutomorphicForm.CuspidalConstituent.iSup_inf_levelInvariantSubmodule_inf_archCutSubmodule_le6 below · cited by 4 · depth 18 - Cuspidal constituents pass to smaller windows
AutomorphicForm.CuspidalConstituent.isCuspConstituent_productionPinsOf_mono0 below · cited by 2 · depth 18 - Twisting a cuspidal constituent by a finite-order Hecke character
AutomorphicForm.CuspidalConstituent.isCuspConstituent_twistedCentralChar_span_image_fnTwist2 below · cited by 2 · depth 18 - Cyclicity: a vector generates a cuspidal sub-representation
AutomorphicForm.CuspidalConstituent.isCuspSubrep_span_cyclic_and_mem_and_le14 below · cited by 4 · depth 18 - Reducing a finite test factor to the level indicator
AutomorphicForm.CuspidalConstituent.isFactorizableTestFn_indicator_and_rightConv_mem_span_rightTranslate_rightConv_indicator_of_mem_levelInvariantSubmodule4 below · cited by 1 · depth 18 - Iterated lowering and raising operators shift the archimedean weight by two
AutomorphicForm.CuspidalConstituent.iterate_lower_mem_cut_ofChar_and_iterate_raise_mem_cut_ofChar164 below · cited by 10 · depth 18 - Window-independence of the K-finite cuspidal space of a constituent
AutomorphicForm.CuspidalConstituent.le_cuspKFiniteSubmodule_of_isCuspConstituent_of_exists_mem_levelInvariantSubmodule112 below · cited by 1 · depth 18 - Non-zero eigenvectors of the isotypic type-cut lie in the cuspidal constituents
AutomorphicForm.CuspidalConstituent.mem_iSup_isCuspConstituent_of_mem_isotypicCuspSubmodule_inf_archCutSubmodule_of_rightConv_eq_smul232 below · cited by 5 · depth 18 - Mean-square approximation by translates forces membership in a constituent
AutomorphicForm.CuspidalConstituent.mem_of_isCuspConstituent_of_mem_of_forall_exists_setLIntegral_ample_sub_sum_mul_translate_sq_lt175 below · cited by 1 · depth 18 - Type-χ vectors of a cyclic span lie in a spherical span
AutomorphicForm.CuspidalConstituent.mem_span_rightTranslate_sup_span_rightConv_spherical_of_mem_span_cyclic_of_mem_archCutSubmodule_ofChar8 below · cited by 1 · depth 18 - Finite-dimensionality of K-invariants of bounded archimedean type
AutomorphicForm.CuspidalConstituent.IsCuspConstituent.finiteDimensional_of_forall_rightTranslate_eq169 below · cited by 1 · depth 19 - Character projector onto the archimedean χ-type
AutomorphicForm.CuspidalConstituent.continuous_and_mem_archCutSubmodule_ofChar_of_eq_integral_rightTranslate_adelicArchGLIncl0 below · cited by 2 · depth 19 - Narrow-window K-finite cusp space sits in the wide-window one
AutomorphicForm.CuspidalConstituent.cuspKFiniteSubmodule_le_cuspKFiniteSubmodule_of_le_of_exists_ne_zero4 below · cited by 2 · depth 19 - Every vector of a cuspidal constituent lies in an archimedean cut
AutomorphicForm.CuspidalConstituent.exists_archTypeFamily_mem_archCutSubmodule_of_mem_isCuspConstituent0 below · cited by 1 · depth 19 - Vectors of level-and-type cuts are right convolutions
AutomorphicForm.CuspidalConstituent.exists_eq_rightConv_of_mem_cut162 below · cited by 10 · depth 19 - A single Casimir eigenvalue on a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_forall_isArchSmoothAt_and_archCasimirAt_eq_smul_of_isCuspConstituent183 below · cited by 11 · depth 19 - Archimedean type cuts inside a cuspidal subrepresentation refine to irreducibles
AutomorphicForm.CuspidalConstituent.exists_inf_archCutSubmodule_le_iSup_isIrreducible_of_isCuspSubrep5 below · cited by 1 · depth 19 - Conjugation preserves archimedean bi-finite factorizable test functions
AutomorphicForm.CuspidalConstituent.exists_isFactorizableTestFn_isArchBiFinite_conj0 below · cited by 1 · depth 19 - Convolution f'*check f of factorizable bi-finite test functions
AutomorphicForm.CuspidalConstituent.exists_isFactorizableTestFn_isArchBiFinite_rightConv_comp_inv8 below · cited by 1 · depth 19 - Factorizable test function reproducing a cuspidal vector
AutomorphicForm.CuspidalConstituent.exists_isFactorizableTestFn_rightConv_eq_self_of_mem_inf_levelInvariantSubmodule_inf_archCutSubmodule165 below · cited by 1 · depth 19 - Non-zero right-convolution eigenvector in a non-zero isotypic cut
AutomorphicForm.CuspidalConstituent.exists_ne_zero_rightConv_eq_smul_of_isotypicCuspSubmodule_inf_archCutSubmodule_ne_bot142 below · cited by 1 · depth 19 - Whittaker decay at the torus origin, finite translate
AutomorphicForm.CuspidalConstituent.exists_norm_whittakerCoefficient_diagOne_mul_le_ideleNorm_rpow_mul_prod_min_of_isCuspConstituent_mul_of_glArch_eq_one290 below · cited by 3 · depth 19 - Level-spherical convolution acts by a real scalar on a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_real_forall_rightConv_eq_smul_of_isLevelSphericalOfType63 below · cited by 2 · depth 19 - Some level-spherical smoothing is injective on a finite-dimensional space
AutomorphicForm.CuspidalConstituent.exists_rightConv_injOn_of_finiteDimensional_of_le21 below · cited by 4 · depth 19 - Independent subfamily with the same sum of cuspidal constituents
AutomorphicForm.CuspidalConstituent.exists_subset_iSupIndep_iSup_eq_of_finset_isCuspConstituent1 below · cited by 2 · depth 19 - χ-averaging a right convolution gives a doubly averaged test factor
AutomorphicForm.CuspidalConstituent.integral_rightConv_rightTranslate_eq_rightConv_doubleAvg3 below · cited by 1 · depth 19 - Smoothing preserves the isotypic cuspidal archimedean cut
AutomorphicForm.CuspidalConstituent.rightConv_mem_isotypicCuspSubmodule_inf_archCutSubmodule83 below · cited by 2 · depth 19 - Right convolution versus right translation on adelic GL₂
AutomorphicForm.CuspidalConstituent.rightConv_rightTranslate_eq_rightTranslate_rightConv_conj1 below · cited by 3 · depth 19 - Finite-adelic translates preserve a cuspidal constituent and its archimedean data
AutomorphicForm.CuspidalConstituent.sum_mul_apply_mul_mem_and_arch_transfer_of_mem_isCuspConstituent_of_mem_finiteAdelicGL2Subgroup0 below · cited by 1 · depth 19 - Casimir at a real place scales a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_forall_isArchSmoothAt_and_archCasimirAt_eq_smul_of_isCuspConstituent_of_exists_isComplex176 below · cited by 1 · depth 20 - Casimir acts by a scalar on a totally real cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_forall_isArchSmoothAt_and_archCasimirAt_eq_smul_of_isCuspConstituent_of_forall_isReal177 below · cited by 1 · depth 20 - Finitely many irreducible archimedean types for simple constituents
AutomorphicForm.CuspidalConstituent.exists_forall_le_archTypeSubmoduleAt_of_isSimple_of_le_iSup0 below · cited by 4 · depth 20 - Uniform bound for archimedean translates of Whittaker coefficients
AutomorphicForm.CuspidalConstituent.exists_forall_whittakerCoefficient_mul_eq_sum_mul_whittakerCoefficient_mul_diagOne_of_isCuspConstituent5 below · cited by 2 · depth 20 - Whittaker decay on the torus for totally real fields
AutomorphicForm.CuspidalConstituent.exists_norm_whittakerCoefficient_diagOne_mul_le_ideleNorm_rpow_mul_prod_min_of_isCuspConstituent_mul_of_glArch_eq_one_of_forall_isReal229 below · cited by 1 · depth 20 - Whittaker torus decay for cuspidal constituents: complex place
AutomorphicForm.CuspidalConstituent.exists_norm_whittakerCoefficient_diagOne_mul_le_ideleNorm_rpow_mul_prod_min_of_isCuspConstituent_mul_of_glArch_eq_one_of_isComplex287 below · cited by 1 · depth 20 - Minimality dichotomy for the level-and-type cut of a cuspidal constituent
AutomorphicForm.CuspidalConstituent.inf_eq_bot_or_le_of_isCuspConstituent34 below · cited by 1 · depth 20 - Archimedean type splitting for stable spaces of continuous functions
AutomorphicForm.CuspidalConstituent.inf_iSup_archTypeSubmoduleAt_le_iSup_inf_of_continuous3 below · cited by 4 · depth 20 - Smoothing by a test function supported in the level group
AutomorphicForm.CuspidalConstituent.rightConv_mem_levelInvariantSubmodule_inf_archCutSubmodule_of_isArchBiFinite3 below · cited by 5 · depth 20 - Stability of weight-one isotypic vectors under reflected lowering
AutomorphicForm.CuspidalConstituent.add_smul_reflect_lower_mem_and_isIsotypicCuspFormAt_of_mem_isCuspConstituent269 below · cited by 1 · depth 21 - Real-place archimedean core hypotheses for pure-weight cut vectors
AutomorphicForm.CuspidalConstituent.coreHypotheses_of_mem_cut_of_forall_hasArchCharacterAt216 below · cited by 2 · depth 21 - Core archimedean hypotheses for pure-weight cut vectors, totally real case
AutomorphicForm.CuspidalConstituent.coreHypotheses_of_mem_cut_of_forall_hasArchCharacterAt_of_forall_isReal211 below · cited by 1 · depth 21 - Weight decomposition of cut vectors at real places
AutomorphicForm.CuspidalConstituent.exists_eq_sum_hasArchCharacterAt_archWeightCharAt_of_isCuspConstituent0 below · cited by 5 · depth 21 - Whittaker coefficients of a cut vector, uniformly over K_∞
AutomorphicForm.CuspidalConstituent.exists_forall_whittakerCoefficient_mul_eq_sum_mul_whittakerCoefficient_mul_diagOne_norm_infinitePlace_eq_one_of_isCuspConstituent5 below · cited by 2 · depth 21 - Pure rotation character in a non-zero level-and-type cut
AutomorphicForm.CuspidalConstituent.exists_inf_archCutSubmodule_ofChar_ne_bot_of_ne_bot1 below · cited by 1 · depth 21 - Irreducible archimedean types cutting U-invariants in a cuspidal subrepresentation
AutomorphicForm.CuspidalConstituent.exists_inf_invariants_le_iSup_isIrreducible_of_isCuspSubrep5 below · cited by 1 · depth 21 - Power bound at a complex place for torus Whittaker coefficients
AutomorphicForm.CuspidalConstituent.exists_norm_whittakerCoefficient_diagOne_mul_le_min_norm_infinitePlace_rpow_of_isComplex_of_glArch_eq_one225 below · cited by 2 · depth 21 - Archimedean decay of torus Whittaker coefficients, two complex places
AutomorphicForm.CuspidalConstituent.exists_norm_whittakerCoefficient_diagOne_mul_le_prod_norm_infinitePlace_rpow_mul_min_rpow_of_forall_hasArchCharacterAt_of_two_le_card_isComplex_of_glArch_eq_one283 below · cited by 1 · depth 21 - A single level-spherical flat test function separating Y
AutomorphicForm.CuspidalConstituent.exists_rightConv_injOn_of_finiteDimensional_of_forall_apply_mul_eq23 below · cited by 1 · depth 21 - Finite-dimensionality and R(J)∘ L-stability of the weight-one slice over ℚ
AutomorphicForm.CuspidalConstituent.finiteDimensional_and_forall_mem_weightOne_slice_of_forall_comp_J_mem_rat168 below · cited by 1 · depth 21 - Finite-dimensional K-span of vectors in finitely many archimedean types
AutomorphicForm.CuspidalConstituent.finiteDimensional_span_rightTranslate_of_mem_iSup_archTypeSubmoduleAt0 below · cited by 9 · depth 21 - Level-and-type cut of the generated cuspidal subrepresentation lies in M
AutomorphicForm.CuspidalConstituent.iInf_isCuspSubrep_inf_levelInvariantSubmodule_inf_archCutSubmodule_le33 below · cited by 1 · depth 21 - Vectors in a level-and-type cut are smooth; the Casimir preserves it
AutomorphicForm.CuspidalConstituent.isArchSmoothAt_and_continuous_archDerivAt_and_archCasimirAt_mem_of_mem_cut173 below · cited by 1 · depth 21 - Casimir stability and smoothness of cut vectors at a real place
AutomorphicForm.CuspidalConstituent.isArchSmoothAt_and_continuous_archDerivAt_and_archCasimirAt_mem_of_mem_cut_ofChar170 below · cited by 1 · depth 21 - Casimir eigenvalues at a complex place: λ'=λ̄, spherical SL₂-invariance
AutomorphicForm.CuspidalConstituent.casimirBar_eq_conj_and_sl2Invariant_of_casimir_eq_zero_of_isCuspConstituent_of_isComplex187 below · cited by 1 · depth 22 - Unitarity constraints on the Casimir eigenvalue at a real place
AutomorphicForm.CuspidalConstituent.casimir_real_and_pos_or_discrete_or_trivial_of_isCuspConstituent197 below · cited by 1 · depth 22 - Casimir trichotomy at a real place for cuspidal constituents
AutomorphicForm.CuspidalConstituent.casimir_real_and_pos_or_discrete_or_trivial_of_isCuspConstituent_of_forall_isReal192 below · cited by 2 · depth 22 - SU(2)-string decomposition of cut vectors at a complex place
AutomorphicForm.CuspidalConstituent.exists_eq_sum_su2String_highestWeight_of_mem_cut_of_isComplex184 below · cited by 2 · depth 22 - Casimir pair acts by scalars on a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_forall_isArchSmoothAtComplex_and_archCasimirAtComplex_eq_smul_of_isCuspConstituent177 below · cited by 2 · depth 22 - Iterated real-place flow derivatives are bounded on determinant shells
AutomorphicForm.CuspidalConstituent.exists_forall_norm_foldr_archDerivAt_le_of_mem_cut174 below · cited by 2 · depth 22 - Finite rank in one complex variable of Whittaker coefficients
AutomorphicForm.CuspidalConstituent.exists_forall_whittakerCoefficient_diagOne_mul_eq_sum_mul_of_isComplex_of_glArch_eq_one206 below · cited by 1 · depth 22 - Raising and lowering operators on a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_iterate_lower_mem_cut_and_iterate_raise_mem_cut_of_hasArchCharacterAt164 below · cited by 5 · depth 22 - Whittaker coefficients of cut cusp vectors are bounded
AutomorphicForm.CuspidalConstituent.exists_norm_whittakerCoefficient_le_mul_ideleNorm_det_rpow_of_isCuspConstituent174 below · cited by 3 · depth 22 - Iterated archimedean derivatives of cut vectors: smoothness and continuity
AutomorphicForm.CuspidalConstituent.isArchSmoothAt_and_continuous_foldr_archDerivAt_of_mem_cut166 below · cited by 8 · depth 22 - Lowering operator and J-translate stay isotypic in a cuspidal constituent
AutomorphicForm.CuspidalConstituent.lower_mem_isotypicCuspSubmodule_and_comp_J_mem_isotypicCuspSubmodule_of_mem3 below · cited by 2 · depth 22 - Casimir bound -(n²+2n)/16 ≤ Reλ at a complex place
AutomorphicForm.CuspidalConstituent.neg_le_casimir_re_of_highestWeight_of_isCuspConstituent_of_isComplex189 below · cited by 1 · depth 22 - Bargmann unitarity inequalities for a weight-n cut vector
AutomorphicForm.CuspidalConstituent.casimir_im_eq_zero_and_nonneg_and_lower_ne_zero_of_mem_cut_ofChar_of_forall_isReal187 below · cited by 1 · depth 23 - Casimir trichotomy at a real place for cuspidal constituents
AutomorphicForm.CuspidalConstituent.casimir_real_and_pos_or_discrete_or_trivial_of_isCuspConstituent_of_exists_isComplex192 below · cited by 1 · depth 23 - Iterated complex flow derivatives bounded on determinant slabs
AutomorphicForm.CuspidalConstituent.exists_forall_norm_foldr_archDerivAtComplex_le_of_mem_cut174 below · cited by 2 · depth 23 - Propagation of SL₂(ℝ)-invariance through a cuspidal constituent
AutomorphicForm.CuspidalConstituent.forall_apply_mul_archRealGLAt_eq_of_isCuspConstituent_of_exists0 below · cited by 2 · depth 23 - Cut words inside M at a compact open level U
AutomorphicForm.CuspidalConstituent.iInf_isCuspSubrep_inf_invariants_inf_archCutSubmodule_le33 below · cited by 1 · depth 23 - Casimir stability of the cut at a complex place
AutomorphicForm.CuspidalConstituent.isArchSmoothAtComplex_and_continuous_archDerivAtComplex_and_archCasimirAtComplex_mem_of_mem_cut174 below · cited by 1 · depth 23 - Iterated complex-place derivatives of cut vectors: smooth and continuous
AutomorphicForm.CuspidalConstituent.isArchSmoothAtComplex_and_continuous_foldr_archDerivAtComplex_of_mem_cut166 below · cited by 3 · depth 23 - Bargmann inequalities at a real place for cut vectors
AutomorphicForm.CuspidalConstituent.casimir_im_eq_zero_and_nonneg_and_lower_ne_zero_of_mem_cut_of_forall_hasArchCharacterAt187 below · cited by 1 · depth 24 - Boundedness of second-order archimedean derivatives on determinant slabs
AutomorphicForm.CuspidalConstituent.exists_forall_norm_archDerivAt_le_of_mem_cut_ofChar_of_forall_isReal178 below · cited by 1 · depth 24 - Slab bound for flow derivatives of a cuspidal cut vector
AutomorphicForm.CuspidalConstituent.exists_forall_norm_archDerivAt_le_of_mem_cut_of_forall_hasArchCharacterAt178 below · cited by 1 · depth 25 - Isotypic cusp forms of principal level split under one smoothing operator
AutomorphicForm.CuspidalConstituent.exists_eq_sum_rightConv_eq_smul_of_mem_isotypicCuspSubmodule_principal_inf_archCutSubmodule326 below · cited by 3 · depth 26 - Eigen-capture of the level-and-type cut at principal level
AutomorphicForm.CuspidalConstituent.exists_inf_levelInvariantSubmodule_principal_inf_archCutSubmodule_le_iSup_rightConv_eq_smul_of_isCuspConstituent89 below · cited by 1 · depth 26 - Isotypic cusp eigenfunctions lie in a sum of cuspidal constituents
AutomorphicForm.CuspidalConstituent.mem_iSup_isCuspConstituent_of_mem_isotypicCuspSubmodule_principal_inf_archCutSubmodule_of_rightConv_eq_smul229 below · cited by 2 · depth 26 - Principal level: splitting archimedean cuts into irreducible types
AutomorphicForm.CuspidalConstituent.exists_inf_archCutSubmodule_le_iSup_isIrreducible_of_isCuspSubrep_principal5 below · cited by 1 · depth 27 - Real scalar action of a flat-symmetric level-spherical convolution
AutomorphicForm.CuspidalConstituent.exists_real_forall_rightConv_eq_smul_of_isLevelSphericalOfType_principal62 below · cited by 1 · depth 27 - A level-spherical test function injective on a finite-dimensional space
AutomorphicForm.CuspidalConstituent.exists_rightConv_injOn_of_finiteDimensional_of_le_principal25 below · cited by 4 · depth 27 - Level-and-type cut distributes over finite sums of cusp subrepresentations
AutomorphicForm.CuspidalConstituent.iSup_inf_levelInvariantSubmodule_principal_inf_archCutSubmodule_le7 below · cited by 1 · depth 27 - Window-independence of K-finite cuspidal constituents at principal level
AutomorphicForm.CuspidalConstituent.le_cuspKFiniteSubmodule_of_isCuspConstituent_of_exists_mem_levelInvariantSubmodule_principal114 below · cited by 1 · depth 27 - Mean-square approximation forces membership in a cuspidal constituent
AutomorphicForm.CuspidalConstituent.mem_of_isCuspConstituent_of_mem_of_forall_exists_setLIntegral_ample_sub_sum_mul_translate_sq_lt_principal177 below · cited by 1 · depth 27 - Right convolution preserves the isotypic cuspidal cut at principal level
AutomorphicForm.CuspidalConstituent.rightConv_mem_isotypicCuspSubmodule_principal_inf_archCutSubmodule83 below · cited by 1 · depth 27 - Casimirs at a complex place act by scalars on a cuspidal constituent, principal level
AutomorphicForm.CuspidalConstituent.exists_forall_isArchSmoothAtComplex_and_archCasimirAtComplex_eq_smul_of_isCuspConstituent_principal179 below · cited by 1 · depth 28 - Casimir at a real place acts by a scalar on a cuspidal constituent
AutomorphicForm.CuspidalConstituent.exists_forall_isArchSmoothAt_and_archCasimirAt_eq_smul_of_isCuspConstituent_principal178 below · cited by 1 · depth 28 - Minimality transfer at principal level for cuspidal constituents
AutomorphicForm.CuspidalConstituent.inf_eq_bot_or_le_of_isCuspConstituent_principal33 below · cited by 1 · depth 28 - Right convolution preserves principal level and archimedean cut
AutomorphicForm.CuspidalConstituent.rightConv_mem_levelInvariantSubmodule_principal_inf_archCutSubmodule_of_isArchBiFinite3 below · cited by 3 · depth 28 - Smoothness and Casimir stability of cut vectors at principal level
AutomorphicForm.CuspidalConstituent.isArchSmoothAtComplex_and_continuous_archDerivAtComplex_and_archCasimirAtComplex_mem_of_mem_cut_principal176 below · cited by 1 · depth 29 - Casimir preserves the level-and-type cut of a cuspidal constituent
AutomorphicForm.CuspidalConstituent.isArchSmoothAt_and_continuous_archDerivAt_and_archCasimirAt_mem_of_mem_cut_principal175 below · cited by 1 · depth 29 - Smoothing vectors of a level-and-type cut, principal level
AutomorphicForm.CuspidalConstituent.exists_eq_rightConv_of_mem_cut_principal164 below · cited by 2 · depth 30
AutomorphicForm.CuspidalSpectrum 65
- From Siegel-window cusp forms to slab fundamental domain members
AutomorphicForm.CuspidalSpectrum.cuspKFiniteSubmodule_le_cuspMemberSubmodule12 below · cited by 18 · depth 19 - Continuous members inject into the weighted L² carrier
AutomorphicForm.CuspidalSpectrum.eq_zero_of_toCarrier_eq_zero3 below · cited by 26 · depth 19 - A continuous nonzero automorphic function pins its central modulus
AutomorphicForm.CuspidalSpectrum.exists_hasModulus_of_isAutomorphicFnAt_of_continuous6 below · cited by 12 · depth 19 - Cuspidal constituents force a modulus on ξ
AutomorphicForm.CuspidalSpectrum.exists_hasModulus_of_isCuspConstituent7 below · cited by 8 · depth 19 - Compact symmetric smoothing operator on the cuspidal spectrum
AutomorphicForm.CuspidalSpectrum.exists_isCompactOperator_isSymmetric_lift_rightConv98 below · cited by 9 · depth 19 - Cuspidal constituent attached to an irreducible closed cusp subrepresentation
AutomorphicForm.CuspidalSpectrum.exists_isCuspConstituent_forall_mem_iff_toCuspSubcarrier_mem_of_isIrreducibleCuspSubrep201 below · cited by 4 · depth 19 - Compact lift of right convolution to the cuspidal sub-carrier
AutomorphicForm.CuspidalSpectrum.exists_isCuspLift_rightConv_isCompactOperator98 below · cited by 2 · depth 19 - Existence of slab fundamental domains for GL₂
AutomorphicForm.CuspidalSpectrum.exists_isSlabFundamentalDomain8 below · cited by 14 · depth 19 - Eigenspace decomposition of right convolution on a finite-dimensional cut
AutomorphicForm.CuspidalSpectrum.exists_le_iSup_rightConv_eq_smul_of_finiteDimensional101 below · cited by 1 · depth 19 - Non-zero convolution eigenvectors come from continuous cusp forms
AutomorphicForm.CuspidalSpectrum.exists_mem_cuspMemberSubmodule_toCuspSubcarrier_eq_rightConv_eq_smul89 below · cited by 5 · depth 19 - Carrier-norm approximation inside a cuspidal subrepresentation from ample-window L² closeness
AutomorphicForm.CuspidalSpectrum.exists_mem_inf_norm_toCuspSubcarrier_sub_lt_of_mem_of_forall_exists_setLIntegral_ample_sub_sum_mul_translate_sq_lt52 below · cited by 1 · depth 19 - Approximate identities for cuspidal vectors in the slab carrier
AutomorphicForm.CuspidalSpectrum.exists_norm_toCarrier_sub_lt41 below · cited by 3 · depth 19 - Eigenvector of a compact cuspidal smoothing operator decomposes discretely
AutomorphicForm.CuspidalSpectrum.exists_orthogonal_isIrreducibleCuspSubrep_sum_eq_of_apply_eq_smul38 below · cited by 4 · depth 19 - Flat-symmetric smoothing non-zero on a cuspidal type cut
AutomorphicForm.CuspidalSpectrum.exists_rightConv_ne_zero_of_ne_bot24 below · cited by 1 · depth 19 - Spectral μ-components of isotypic cusp forms are eigenfunction classes
AutomorphicForm.CuspidalSpectrum.exists_slice_sub_mem_eigenspace_orthogonal120 below · cited by 2 · depth 19 - Isotypic cusp forms persist under projection to a closed subrepresentation
AutomorphicForm.CuspidalSpectrum.isIsotypicCuspFormAt_of_mem_of_sub_mem_orthogonal43 below · cited by 1 · depth 19 - Isotypic cusp forms lie in the slab cusp-member submodule
AutomorphicForm.CuspidalSpectrum.isotypicCuspSubmodule_le_cuspMemberSubmodule12 below · cited by 3 · depth 19 - Convolution eigenfunctions among cuspidal members are K-finite
AutomorphicForm.CuspidalSpectrum.mem_cuspKFiniteSubmodule_of_mem_cuspMemberSubmodule_of_rightConv_eq_smul83 below · cited by 2 · depth 19 - Right convolution preserves cuspidal continuous members
AutomorphicForm.CuspidalSpectrum.rightConv_mem_cuspMemberSubmodule22 below · cited by 13 · depth 19 - Right translation preserves the cuspidal member submodule
AutomorphicForm.CuspidalSpectrum.rightTranslate_mem_cuspMemberSubmodule15 below · cited by 23 · depth 19 - Lifted right convolution preserves the cuspidal subcarrier
AutomorphicForm.CuspidalSpectrum.apply_mem_cuspSubcarrier_of_isLift_rightConv7 below · cited by 7 · depth 20 - Hecke coset sums lift to operators commuting with smoothing
AutomorphicForm.CuspidalSpectrum.exists_commute_lift_heckeCosetSum_of_isLevelSphericalOfType37 below · cited by 1 · depth 20 - Slab square mass dominated by ample Siegel window mass
AutomorphicForm.CuspidalSpectrum.exists_forall_setLIntegral_le_mul_setLIntegral_of_isSlabFundamentalDomain_of_coversModCentre_ample11 below · cited by 2 · depth 20 - Existence of a level-and-type cut projector on the cuspidal carrier
AutomorphicForm.CuspidalSpectrum.exists_idempotent_cutProjector_of_isCompact43 below · cited by 4 · depth 20 - Right convolution lifts to an adjoint pair of operators
AutomorphicForm.CuspidalSpectrum.exists_isLift_rightConv15 below · cited by 13 · depth 20 - Right translation lifts to bounded operators on the weighted L² carrier
AutomorphicForm.CuspidalSpectrum.exists_isLift_rightTranslate10 below · cited by 9 · depth 20 - Non-zero closed cuspidal subrepresentations contain K-finite vectors
AutomorphicForm.CuspidalSpectrum.exists_ne_zero_mem_cuspKFiniteSubmodule_toCuspSubcarrier_mem_of_isClosedCuspSubrep_of_ne_bot138 below · cited by 3 · depth 20 - Strong continuity of right translation on the weighted L² carrier
AutomorphicForm.CuspidalSpectrum.exists_nhds_forall_norm_toCarrier_rightTranslate_sub_lt14 below · cited by 5 · depth 20 - Admissibility of irreducible closed cuspidal subrepresentations, at function level
AutomorphicForm.CuspidalSpectrum.finiteDimensional_of_le_cuspKFiniteSubmodule_of_toCuspSubcarrier_mem_of_isIrreducibleCuspSubrep186 below · cited by 1 · depth 20 - Orthogonal complement of a closed cuspidal subrepresentation
AutomorphicForm.CuspidalSpectrum.isClosedCuspSubrep_orthogonal36 below · cited by 3 · depth 20 - Closure of a cuspidal subrepresentation's class image is closed subrepresentation
AutomorphicForm.CuspidalSpectrum.isClosedCuspSubrep_topologicalClosure_map_toCuspSubcarrier_of_isCuspSubrep0 below · cited by 4 · depth 20 - Compactness of right convolution on the cuspidal subspace
AutomorphicForm.CuspidalSpectrum.isCompactOperator_lift_rightConv_comp_cuspSubcarrier86 below · cited by 2 · depth 20 - K-finite cuspidal preimage of a closed cuspidal sub-representation
AutomorphicForm.CuspidalSpectrum.isCuspSubrep_cuspKFiniteSubmodule_inf_map_subtype_comap_toCuspSubcarrier_of_isClosedCuspSubrep97 below · cited by 2 · depth 20 - Flat involution preserves factorizable test functions on GL₂(A_F)
AutomorphicForm.CuspidalSpectrum.isFactorizableTestFn_flat5 below · cited by 3 · depth 20 - Right U(N)-invariant smoothing kills the level-N orthogonal complement
AutomorphicForm.CuspidalSpectrum.apply_eq_zero_of_mem_orthogonal_cuspLevelSubcarrier_of_isLift_rightConv_of_rightInvariant20 below · cited by 1 · depth 21 - An idempotent archimedean type projector on the cuspidal carrier
AutomorphicForm.CuspidalSpectrum.exists_idempotent_archTypeProjector36 below · cited by 1 · depth 21 - A level-average idempotent on the cuspidal carrier
AutomorphicForm.CuspidalSpectrum.exists_idempotent_levelAverage_of_isCompact29 below · cited by 1 · depth 21 - Archimedean bi-finiteness of the flat of a test function
AutomorphicForm.CuspidalSpectrum.exists_isArchBiFinite_flat0 below · cited by 1 · depth 21 - Flat bi-finite smoothings separate vectors of the cuspidal sub-carrier
AutomorphicForm.CuspidalSpectrum.exists_isArchBiFinite_flat_isCompactOperator_lift_rightConv_apply_ne_zero130 below · cited by 3 · depth 21 - Right translation on the cuspidal sub-carrier, norm ≤‖det y‖^{σ/2}
AutomorphicForm.CuspidalSpectrum.exists_isCuspLift_rightTranslate_and_norm_le17 below · cited by 4 · depth 21 - Right convolution is bounded on the weighted L² carrier
AutomorphicForm.CuspidalSpectrum.exists_norm_toCarrier_rightConv_le13 below · cited by 1 · depth 21 - Flat involution preserves archimedean test factors
AutomorphicForm.CuspidalSpectrum.isArchTestFactor_conj_inv_mul_ideleNorm_det_rpow3 below · cited by 1 · depth 21 - Flat involution preserves finite test factors
AutomorphicForm.CuspidalSpectrum.isFinTestFactor_conj_inv_mul_ideleNorm_det_rpow3 below · cited by 1 · depth 21 - Smoothing maps the cuspidal subcarrier into its level-N part
AutomorphicForm.CuspidalSpectrum.map_cuspSubcarrier_le_cuspLevelSubcarrier_of_isLift_rightConv8 below · cited by 1 · depth 21 - Spectral dichotomy for a typed level cut in an irreducible cuspidal subrepresentation
AutomorphicForm.CuspidalSpectrum.map_inf_orthogonal_eq_bot_or_le_of_isIrreducibleCuspSubrep94 below · cited by 1 · depth 21 - Smoothed cuspidal member is K-finite at Siegel pins
AutomorphicForm.CuspidalSpectrum.rightConv_mem_cuspKFiniteSubmodule_of_mem_cuspMemberSubmodule_of_isArchBiFinite91 below · cited by 2 · depth 21 - Hecke coset-sum operator on the cuspidal sub-carrier, compact level
AutomorphicForm.CuspidalSpectrum.exists_commute_lift_cosetSum_of_isLevelSphericalOfType_of_isCompact32 below · cited by 2 · depth 22 - Commuting bounded lift of right convolution on the cuspidal subcarrier
AutomorphicForm.CuspidalSpectrum.exists_commute_lift_rightConv_of_isArchBiFinite_of_isCompact23 below · cited by 1 · depth 22 - Translation lift commuting with a level-spherical smoothing operator
AutomorphicForm.CuspidalSpectrum.exists_commute_lift_rightTranslate_rowIsometry_of_isCompact30 below · cited by 1 · depth 22 - Square mass on a slab fundamental domain dominated by a covering Siegel window
AutomorphicForm.CuspidalSpectrum.exists_forall_setLIntegral_le_mul_setLIntegral_of_isSlabFundamentalDomain_of_coversModCentre11 below · cited by 1 · depth 22 - Flat level-U spherical approximate identity in the cuspidal subcarrier
AutomorphicForm.CuspidalSpectrum.exists_isLevelSphericalOfType_flat_norm_toCuspSubcarrier_sub_lt_of_forall_apply_mul_eq41 below · cited by 2 · depth 22 - Non-zero cusp-carrier vectors pair with K-finite level forms
AutomorphicForm.CuspidalSpectrum.exists_mem_archCutSubmodule_inner_toCuspSubcarrier_ne_zero_of_ne_zero32 below · cited by 1 · depth 22 - Isometric strongly continuous representation on the cuspidal subcarrier
AutomorphicForm.CuspidalSpectrum.exists_monoidHom_isCuspLift_rightTranslate_and_norm_eq_and_continuous24 below · cited by 2 · depth 22 - Strongly continuous isometric U-action on the cuspidal carrier
AutomorphicForm.CuspidalSpectrum.exists_monoidHom_isCuspLift_rightTranslate_coe_and_norm_eq_and_continuous_of_isCompact23 below · cited by 1 · depth 22 - Lifted averages over compact finite-adelic subgroups are averages of translates
AutomorphicForm.CuspidalSpectrum.integral_smul_apply_toCuspSubcarrier_eq_toCuspSubcarrier_integral_mul_apply_mul_of_isCompact19 below · cited by 1 · depth 22 - Dichotomy for classes of a typed level cut in L
AutomorphicForm.CuspidalSpectrum.map_inf_eq_bot_or_le_of_isIrreducibleCuspSubrep_of_isClosed84 below · cited by 1 · depth 22 - Right translation by determinant-norm-one elements is unitary
AutomorphicForm.CuspidalSpectrum.rightTranslate_mem_and_pairing_rightTranslate_eq_of_ideleNorm_det_eq_one7 below · cited by 8 · depth 22 - Lifted averages of translates represent function-level averages
AutomorphicForm.CuspidalSpectrum.integral_smul_apply_toCuspSubcarrier_eq_toCuspSubcarrier_integral_mul_rightTranslate19 below · cited by 1 · depth 23 - K-finite cusp functions with classes in a closed cuspidal subrepresentation
AutomorphicForm.CuspidalSpectrum.isCuspSubrep_cuspKFiniteSubmodule_fdPins_inf_map_subtype_comap_toCuspSubcarrier_of_isClosedCuspSubrep33 below · cited by 1 · depth 23 - Finite eigenspace decomposition of a principal-level cuspidal cut
AutomorphicForm.CuspidalSpectrum.exists_le_iSup_rightConv_eq_smul_of_finiteDimensional_principal101 below · cited by 1 · depth 27 - Principal-level spherical flat test functions approximate cuspidal L² classes
AutomorphicForm.CuspidalSpectrum.exists_norm_toCarrier_sub_lt_principal41 below · cited by 1 · depth 27 - Flat level-spherical smoothing non-vanishing on a type cut
AutomorphicForm.CuspidalSpectrum.exists_rightConv_ne_zero_of_ne_bot_principal28 below · cited by 1 · depth 27 - Isotypic cuspidal slice vector for a μ-eigenvalue, principal level
AutomorphicForm.CuspidalSpectrum.exists_slice_sub_mem_eigenspace_orthogonal_principal121 below · cited by 1 · depth 27 - Orthogonal component of an isotypic cusp form at principal level
AutomorphicForm.CuspidalSpectrum.isIsotypicCuspFormAt_principal_of_mem_of_sub_mem_orthogonal43 below · cited by 1 · depth 27 - Carrier approximation by typed level-N vectors in a cuspidal subrepresentation
AutomorphicForm.CuspidalSpectrum.exists_mem_inf_norm_toCuspSubcarrier_sub_lt_of_mem_of_forall_exists_setLIntegral_ample_sub_sum_mul_translate_sq_lt_principal52 below · cited by 1 · depth 28
AutomorphicForm.CyclicBaseChangeLifting 2
- Cubic base change: eigensystems differ by a ray-class twist
AutomorphicForm.CyclicBaseChangeLifting.exists_rayClassChar_twist_of_isBaseChangeOf_of_isArithGenuineCuspRealizable_of_finrank_eq_three890 below · cited by 1 · depth 15 - Quadratic base change: a common lift forces a ray-class twist
AutomorphicForm.CyclicBaseChangeLifting.exists_rayClassChar_twist_of_isBaseChangeOf_of_isArithGenuineCuspRealizable_of_finrank_eq_two890 below · cited by 1 · depth 16
AutomorphicForm.GL2Real 22
- Normal forms and norm fibres in GL₂(ℝ)
AutomorphicForm.GL2Real.exists_conj_normalForm_and_normFibre_and_nonNorm_conjAe0 below · cited by 1 · depth 22 - Bi-finite functions on GL₂(ℝ) with prescribed orbital transforms, in families
AutomorphicForm.GL2Real.exists_contDiff_splitTransform_eq_ellipticTransform_eq_of_discreteSeriesPairing11 below · cited by 1 · depth 22 - Orbital integrals at split and elliptic elements of GL₂(ℝ)
AutomorphicForm.GL2Real.orbitalIntegral_eq_splitTransform_div_and_eq_ellipticTransform_div6 below · cited by 3 · depth 22 - Smoothness and support of the Chebyshev modes of the elliptic transform
AutomorphicForm.GL2Real.contDiff_integral_ellipticTransform_entrySlice_mul_chebyshevU1 below · cited by 1 · depth 23 - Smoothness and symmetry of split transforms in families
AutomorphicForm.GL2Real.contDiff_splitTransform_entrySlice0 below · cited by 1 · depth 23 - Vanishing of discrete-series pairings for rotation type m
AutomorphicForm.GL2Real.discreteSeriesPairing_entrySlice_eq_zero_of_weight0 below · cited by 1 · depth 23 - Continuity and |sinθ| bound for elliptic transforms
AutomorphicForm.GL2Real.ellipticTransform_entrySlice_continuousOn_and_exists_norm_le_mul_abs_sin0 below · cited by 1 · depth 23 - Low Chebyshev modes of the elliptic transform on GL₂(ℝ)
AutomorphicForm.GL2Real.exists_intervalIntegral_ellipticTransform_mul_chebyshevU_eq_of_le_weight0 below · cited by 1 · depth 23 - Weight-one linear inverse of the split transform, in families
AutomorphicForm.GL2Real.exists_linear_entrySlice_archWeightChar_one_splitTransform_eq3 below · cited by 1 · depth 23 - A linear right inverse of the split transform at weight zero
AutomorphicForm.GL2Real.exists_linear_entrySlice_archWeightChar_zero_splitTransform_eq2 below · cited by 2 · depth 23 - Haar measure on an elliptic torus in polar form
AutomorphicForm.GL2Real.exists_map_val_centralizer_ellipticElt_eq_smul_map_ellipticElt0 below · cited by 2 · depth 23 - Haar measure on a split maximal torus of GL₂(ℝ)
AutomorphicForm.GL2Real.exists_map_val_centralizer_upperTriangular_eq_smul_map_diag0 below · cited by 2 · depth 23 - Elliptic product chart: density ρ³/y⁴ pushed to Lebesgue measure
AutomorphicForm.GL2Real.map_ellipticProduct0 below · cited by 1 · depth 23 - Elliptic chart: density ρ³/y⁴ carried to negative determinant
AutomorphicForm.GL2Real.map_ellipticProduct_neg0 below · cited by 1 · depth 23 - Split product chart carries |a₁a₂| dλ to Lebesgue measure
AutomorphicForm.GL2Real.map_splitProduct0 below · cited by 1 · depth 23 - Elliptic coordinates on GL₂(ℝ): change of variables
AutomorphicForm.GL2Real.setIntegral_image_ellipticCoords_eq_integral_jacobian_smul0 below · cited by 1 · depth 24 - Harish-Chandra jump formula at the elliptic torus of GL₂(ℝ)
AutomorphicForm.GL2Real.exists_ne_zero_tendsto_ellipticTransform_entrySlice_div_sin_sub_div_nhdsWithin_Ioi3 below · cited by 3 · depth 30 - The elliptic transform jump constant equals -8π
AutomorphicForm.GL2Real.eq_neg_eight_mul_pi_of_forall_tendsto_ellipticTransform_entrySlice3 below · cited by 1 · depth 31 - Elliptic transform near a positive scalar: unipotent orbital limit
AutomorphicForm.GL2Real.tendsto_ellipticTransform_div_two_mul_sin_nhdsWithin_Ioi_zero0 below · cited by 2 · depth 31 - Vanishing of the first term in the elliptic jump relation
AutomorphicForm.GL2Real.tendsto_sin_mul_integral_fderiv_entrySlice_one_div_nhdsWithin_Ioi_zero0 below · cited by 2 · depth 31 - Poisson-kernel term concentrates at the scalar r· 1
AutomorphicForm.GL2Real.tendsto_sin_mul_integral_fderiv_entrySlice_sub_div_nhdsWithin_Ioi_zero0 below · cited by 2 · depth 31 - Haar mass 2π of the Iwasawa box in GL₂(ℝ)
AutomorphicForm.GL2Real.withDensity_volume_iwasawaBox_eq_two_mul_pi0 below · cited by 1 · depth 33
AutomorphicForm.GL2Twisted 12
- Smoothness of the Chebyshev modes of the twisted elliptic transform
AutomorphicForm.GL2Twisted.contDiff_integral_twistedEllipticTransform_mul_chebyshevU0 below · cited by 1 · depth 22 - Smoothness and symmetry of the twisted split transform
AutomorphicForm.GL2Twisted.contDiff_twistedSplitTransform0 below · cited by 1 · depth 22 - Vanishing of large-weight discrete-series pairings of twisted transforms
AutomorphicForm.GL2Twisted.exists_forall_discreteSeriesPairing_twistedSplitTransform_twistedEllipticTransform_eq_zero3 below · cited by 1 · depth 22 - Continuity, |sinθ| bound and support of twisted elliptic transforms
AutomorphicForm.GL2Twisted.twistedEllipticTransform_continuousOn_and_exists_norm_le_mul_abs_sin0 below · cited by 1 · depth 22 - Twisted orbital integrals on GL₂(ℂ): split and elliptic cases
AutomorphicForm.GL2Twisted.twistedOrbitalIntegral_eq_twistedSplitTransform_div_and_eq_twistedEllipticTransform_div4 below · cited by 1 · depth 22 - Elliptic and split fibre pairings agree in bidegree (0,0)
AutomorphicForm.GL2Twisted.ellipticFibreTerm_pairing_eq_splitFibreTerm_pairing0 below · cited by 1 · depth 23 - Haar measure on GL₂(ℂ⊗_ℝℝ) as |det|⁻⁴ Lebesgue measure
AutomorphicForm.GL2Twisted.exists_isHaarMeasure_eq_smul_map_normSq_det_sq_inv0 below · cited by 2 · depth 23 - Complex split product chart pushes a density to Lebesgue measure
AutomorphicForm.GL2Twisted.map_splitProductChart0 below · cited by 2 · depth 23 - Twisted transforms of a monomial input equal fibre sides
AutomorphicForm.GL2Twisted.twistedTransforms_monomialInput_eq_fibreSides0 below · cited by 1 · depth 23 - Bi-invariance of the unitary chart average on GL₂(ℂ)
AutomorphicForm.GL2Twisted.unitaryAverage_translate0 below · cited by 2 · depth 23 - Iwasawa coordinate integration formula on GL₂(ℂ)
AutomorphicForm.GL2Twisted.exists_pos_forall_integral_eq_mul_setIntegral_iwasawaChart2 below · cited by 1 · depth 32 - Hopf-coordinate average over U(2) as an integral over ψ∈(0,π)
AutomorphicForm.GL2Twisted.unitaryAverage_eq_mul_setIntegral_of_continuous0 below · cited by 1 · depth 32
AutomorphicForm.HeckeEigensystem 7
- Fibre of quadratic base change at Siegel windows
AutomorphicForm.HeckeEigensystem.agreesAwayFromFinite_or_twist_of_formalBaseChange_agreesAwayFromFinite_of_finrank_eq_two_of_coversModCentre897 below · cited by 1 · depth 14 - Cubic base-change fibre: twist by a Galois character
AutomorphicForm.HeckeEigensystem.exists_char_twist_artinFrob_of_formalBaseChange_agreesAwayFromFinite_of_finrank_eq_three_of_coversModCentre_of_pos898 below · cited by 1 · depth 14 - Quadratic base-change fibre: agreement or quadratic twist
AutomorphicForm.HeckeEigensystem.agreesAwayFromFinite_or_twist_of_formalBaseChange_agreesAwayFromFinite_of_finrank_eq_two_of_coversModCentre_of_pos896 below · cited by 1 · depth 15 - Twist relation for two eigensystems with a common base change
AutomorphicForm.HeckeEigensystem.exists_pow_twist_of_isBaseChangeOf_of_isArithGenuineCuspRealizable772 below · cited by 2 · depth 16 - Fibre identity for base-change Rankin–Selberg Euler products
AutomorphicForm.HeckeEigensystem.hasProd_rsEulerPoly_contragredient_fibre_eq_prod_twist_of_isBaseChangeOf0 below · cited by 1 · depth 17 - Conjugates of a,b when ā b=c̄ a and ‖b‖=c
AutomorphicForm.HeckeEigensystem.conj_eq_mul_div_and_conj_eq_sq_mul_inv_of_mul_conj_eq_of_norm_eq0 below · cited by 1 · depth 24 - Galois-conjugate primes have equal absolute norm
AutomorphicForm.HeckeEigensystem.cNorm_eq_of_asIdeal_eq_smul0 below · cited by 3 · depth 27
AutomorphicForm.IdeleChar 1
- A finite-order Hecke character of ℚ of modulus (3)
AutomorphicForm.IdeleChar.exists_finiteOrderHeckeChar_chiNegThree1 below · cited by 1 · depth 14
AutomorphicForm.IsArchTestFactor 2
- Archimedean test factors descend to smooth functions on invertible matrices
AutomorphicForm.IsArchTestFactor.exists_contDiff_hasCompactSupport_tsupport_subset_isUnit_det0 below · cited by 2 · depth 22 - Domination of an archimedean test factor by a nonnegative one
AutomorphicForm.IsArchTestFactor.exists_isArchTestFactor_nonneg_norm_le2 below · cited by 1 · depth 33
AutomorphicForm.IsCuspidalFn 3
- Additivity of cuspidality for integrable constant-term integrands
AutomorphicForm.IsCuspidalFn.add1 below · cited by 2 · depth 19 - Cuspidality is preserved by scalar multiplication
AutomorphicForm.IsCuspidalFn.smul1 below · cited by 2 · depth 19 - Cuspidality is preserved by right translation
AutomorphicForm.IsCuspidalFn.rightTranslate1 below · cited by 1 · depth 20
AutomorphicForm.IsFactorizableTestFn 3
- Unipotent slices of factorizable test functions are pure tensors
AutomorphicForm.IsFactorizableTestFn.comp_mul_unipotentGL2_mul_mem_pureTensorSet0 below · cited by 4 · depth 17 - Factorisable test functions are smooth and differentiable at a real place
AutomorphicForm.IsFactorizableTestFn.isArchSmoothAt_and_archDerivAt_eq_tensor0 below · cited by 5 · depth 20 - Factorizable test functions: smoothness and tensor flow derivatives at a complex place
AutomorphicForm.IsFactorizableTestFn.isArchSmoothAtComplex_and_archDerivAtComplex_eq_tensor0 below · cited by 4 · depth 25
AutomorphicForm.IsFinTestFactor 1
- Compact open right-invariance subgroup for finite test factors
AutomorphicForm.IsFinTestFactor.exists_isCompact_isOpen_forall_mul_eq0 below · cited by 2 · depth 24
AutomorphicForm.IsGL2RealKTypeModule 1
- K-type support of an irreducible infinite-dimensional GL₂(ℝ)-module
AutomorphicForm.IsGL2RealKTypeModule.ne_bot_iff_parity_or_discreteSeries_of_irreducible0 below · cited by 1 · depth 17
AutomorphicForm.IsInducedSection 2
- Induced sections are left invariant under B(F) and N(A_F)
AutomorphicForm.IsInducedSection.apply_globalPoints_mul_of_mem_borelSubgroup_and_apply_unipotentGL2_mul3 below · cited by 10 · depth 20 - Induced sections agreeing on the maximal compact are equal
AutomorphicForm.IsInducedSection.eq_of_eqOn_maximalCompact2 below · cited by 5 · depth 20
AutomorphicForm.IsIsotypicCuspFormAt 2
- Monotonicity of isotypic cusp forms in level and bad set
AutomorphicForm.IsIsotypicCuspFormAt.of_le_of_subset2 below · cited by 5 · depth 17 - Isotypic cusp forms as smooth-cusp realizations at level N
AutomorphicForm.IsIsotypicCuspFormAt.exists_smoothCuspRealizationAt_toFun_eq_of_ne_bot0 below · cited by 10 · depth 18
AutomorphicForm.IsKfSmooth 3
- K_f-smooth functions admit a level of unipotent invariance
AutomorphicForm.IsKfSmooth.exists_ideal_forall_apply_mul_conj_unipotentGL2_eq0 below · cited by 4 · depth 15 - K_f-smooth functions are right-invariant under a compact subgroup
AutomorphicForm.IsKfSmooth.exists_isCompact_isOpen_eq_inf_forall_apply_mul_eq1 below · cited by 1 · depth 20 - Finitely many translates of a K_f-smooth function on a compact set
AutomorphicForm.IsKfSmooth.finite_smul_image_of_isCompact0 below · cited by 4 · depth 22
AutomorphicForm.IsOrbitalIntegralOn 2
- Well-definedness of the orbital integral at a regular semisimple element
AutomorphicForm.IsOrbitalIntegralOn.unique_of_isRegularSemisimple1 below · cited by 19 · depth 19 - Existence of adelic orbital integrals at regular semisimple γ
AutomorphicForm.IsOrbitalIntegralOn.exists_adeleRing_of_isRegularSemisimple4 below · cited by 1 · depth 27
AutomorphicForm.IsRegularSemisimple 1
- Centralisers of regular semisimple elements in GL₂ commute
AutomorphicForm.IsRegularSemisimple.mul_comm_of_mem_centralizer0 below · cited by 1 · depth 27
AutomorphicForm.IsSlabProfile 2
- Right convolution by a test function preserves slab profiles
AutomorphicForm.IsSlabProfile.convOp5 below · cited by 1 · depth 35 - Triviality of a slab profile's central character on principal ideles
AutomorphicForm.IsSlabProfile.apply_eq_one_of_mem_principalIdeles_of_apply_ne_zero0 below · cited by 1 · depth 36
AutomorphicForm.IsTwistedOrbitalIntegralOn 2
- Uniqueness of the twisted orbital integral at δ with regular semisimple norm
AutomorphicForm.IsTwistedOrbitalIntegralOn.unique_of_isRegularSemisimple_normString1 below · cited by 14 · depth 19 - Existence of twisted orbital integrals at regular semisimple norms
AutomorphicForm.IsTwistedOrbitalIntegralOn.exists_of_isRegularSemisimple_normString_of_finrank_eq_two0 below · cited by 1 · depth 33
AutomorphicForm.IsTwistedWeightedOrbitalIntegralOn 1
- Section-function independence of twisted weighted orbital integrals
AutomorphicForm.IsTwistedWeightedOrbitalIntegralOn.unique_of_isRegularSemisimple_normString_of_forall_twistedCentralizer_mul_eq1 below · cited by 3 · depth 33
AutomorphicForm.IsUnitFactorizableAbove 1
- Support of unit-factorizable functions is integral outside S
AutomorphicForm.IsUnitFactorizableAbove.finComponent_glFin_mem_localIntegralSet_of_apply_ne_zero0 below · cited by 1 · depth 21
AutomorphicForm.IsWeightedOrbitalIntegralOn 1
- Uniqueness of the weighted orbital integral of a regular semisimple element
AutomorphicForm.IsWeightedOrbitalIntegralOn.unique_of_isRegularSemisimple_of_forall_centralizer_mul_eq1 below · cited by 3 · depth 32
AutomorphicForm.LocalFunctionSpace 8
- Vanishing at 1 forces membership in the twisted unipotent span
AutomorphicForm.LocalFunctionSpace.mem_span_sub_of_apply_one_eq_zero_of_irreducible_of_admissible7 below · cited by 5 · depth 18 - Vanishing of a Whittaker function with trivial Kirillov image
AutomorphicForm.LocalFunctionSpace.eq_zero_of_forall_diagonal_mul_mem_span_sub2 below · cited by 1 · depth 19 - Finite Fourier expansion of a locally constant function on a p-adic ball
AutomorphicForm.LocalFunctionSpace.exists_finset_forall_eq_sum_mul_char_mul0 below · cited by 2 · depth 19 - Cutting off a GL₂ function by a ball indicator modulo twisted defects
AutomorphicForm.LocalFunctionSpace.exists_mem_forall_diagonal_mul_sub_mem_span_and_mem_span0 below · cited by 1 · depth 19 - Schur's lemma for an irreducible admissible function space on GL₂
AutomorphicForm.LocalFunctionSpace.exists_smul_eq_of_irreducible_of_admissible0 below · cited by 1 · depth 19 - Constancy from vanishing of all twisted finite window sums
AutomorphicForm.LocalFunctionSpace.eq_of_forall_exists_forall_sum_char_mul_eq_zero0 below · cited by 1 · depth 20 - Vanishing of unipotent-invariant local Whittaker functions
AutomorphicForm.LocalFunctionSpace.eq_zero_of_forall_mul_unipotent_eq0 below · cited by 1 · depth 20 - Injectivity of the Kirillov map on a Whittaker space
AutomorphicForm.LocalFunctionSpace.eq_zero_of_forall_apply_diagOne_eq_zero_of_irreducible_of_admissible8 below · cited by 2 · depth 22
AutomorphicForm.LocalIntertwining 19
- Adelic factorisation of an unramified intertwining integral
AutomorphicForm.LocalIntertwining.integral_adeleRing_pureTensor_prod_mul_finprod_unramifiedWeylIntegrand_mul_tprod5 below · cited by 9 · depth 21 - Beta integral on ℂ for zᵃ̄ z^{ b}(1+|z|²)^{-t}
AutomorphicForm.LocalIntertwining.integral_pow_mul_conj_pow_mul_one_add_norm_sq_cpow_neg1 below · cited by 6 · depth 21 - Shell-by-shell local intertwining integral at a finite place
AutomorphicForm.LocalIntertwining.integral_smoothWeylIntegrand_adicCompletion0 below · cited by 9 · depth 21 - Weight-k archimedean intertwining integral as a Γ_ℝ-quotient
AutomorphicForm.LocalIntertwining.integral_sub_I_div_sqrt_one_add_sq_zpow_mul_cpow_neg_eq_GammaReal1 below · cited by 6 · depth 21 - Finite-adelic unramified intertwining integral as an Euler product
AutomorphicForm.LocalIntertwining.integral_finiteAdeleRing_prod_mul_finprod_unramifiedWeylIntegrand_mul_tprod2 below · cited by 1 · depth 22 - Area integral of (1+|z|²)^{-(2s+1)} over ℂ
AutomorphicForm.LocalIntertwining.integral_one_add_norm_sq_cpow_neg_eq_pi_div0 below · cited by 1 · depth 22 - Euler's beta integral int_ℝ(1+x²)^{-(s+1/2)}=Γ_ℝ(2s)/Γ_ℝ(2s+1)
AutomorphicForm.LocalIntertwining.integral_one_add_sq_cpow_neg_eq_GammaReal_div0 below · cited by 1 · depth 22 - Unramified rank-one intertwining integral at a finite place
AutomorphicForm.LocalIntertwining.integral_unramifiedWeylIntegrand_adicCompletion0 below · cited by 7 · depth 22 - Complex-place atom: uniform bound and Möbius-shift limit
AutomorphicForm.LocalIntertwining.bounded_and_tendsto_integral_moebiusShift_sub_integral_complexAtom2 below · cited by 1 · depth 26 - Uniform bound and Möbius-shift limit for the real atom
AutomorphicForm.LocalIntertwining.bounded_and_tendsto_integral_moebiusShift_sub_integral_realAtom2 below · cited by 1 · depth 26 - Boundedness and vanishing limit for the local intertwining atom at σdownarrow 1/2
AutomorphicForm.LocalIntertwining.bounded_and_tendsto_integral_weylShift_sub_integral_smoothAtom_adicCompletion4 below · cited by 2 · depth 26 - Uniform bound for archimedean K-type intertwining integrals
AutomorphicForm.LocalIntertwining.exists_bound_norm_integral_mixedSpace_archAtom_prod0 below · cited by 1 · depth 26 - Integrability of the complex-place atom and its Möbius translate
AutomorphicForm.LocalIntertwining.integrable_complexAtom_and_integrable_moebiusShift0 below · cited by 1 · depth 26 - Integrability of the real-place intertwining atom and its Möbius shift
AutomorphicForm.LocalIntertwining.integrable_realAtom_and_integrable_moebiusShift0 below · cited by 1 · depth 26 - Integrability of a local intertwining atom and its Weyl translate
AutomorphicForm.LocalIntertwining.integrable_smoothAtom_and_integrable_weylShift_adicCompletion3 below · cited by 2 · depth 26 - Vanishing of the twisted complex atom integral as σdownarrow 1/2
AutomorphicForm.LocalIntertwining.tendsto_integral_normSq_rpow_sub_one_mul_complexAtom_nhdsGT_one_half0 below · cited by 1 · depth 27 - Vanishing of the Möbius weight correction as σdownarrow 1/2
AutomorphicForm.LocalIntertwining.tendsto_integral_sq_rpow_sub_one_mul_realAtom_nhdsGT_one_half0 below · cited by 1 · depth 27 - Polynomiality of the local intertwining integral in N(v)^{-2s}
AutomorphicForm.LocalIntertwining.exists_one_sub_mul_integral_smoothWeylIntegrand_eq_sum3 below · cited by 4 · depth 30 - Meromorphic continuation of a local intertwining integral
AutomorphicForm.LocalIntertwining.exists_meromorphicOn_eq_integral_smoothWeylIntegrand_adicCompletion1 below · cited by 1 · depth 32
AutomorphicForm.LocalWeightedOrbital 6
- Half-weight along diag(a,at): compact support, local constancy off t=1
AutomorphicForm.LocalWeightedOrbital.exists_isCompact_forall_halfWeighted_ne_zero_mem_and_forall_exists_nhds_halfWeighted_eq_of_isLocalTestFn2 below · cited by 1 · depth 35 - Local constancy of the half-weighted orbital integral near t=1
AutomorphicForm.LocalWeightedOrbital.exists_nhds_forall_halfWeighted_mul_eq_halfWeighted_mul_of_norm_sub_le_of_isLocalTestFn2 below · cited by 2 · depth 35 - Weighted orbital integral at diag(a,b) versus Langlands' half-weight
AutomorphicForm.LocalWeightedOrbital.ratio_mul_sqrtRatio_mul_eq_neg_two_mul_halfWeighted_of_isWeightedOrbitalIntegral5 below · cited by 2 · depth 35 - Logarithmic expansion of half-weighted orbital integrals near t=1
AutomorphicForm.LocalWeightedOrbital.exists_forall_splitOrbital_eq_and_norm_two_mul_halfWeighted_sub_le_of_isLocalTestFn3 below · cited by 1 · depth 38 - Split orbital vanishing at a non-norm parameter
AutomorphicForm.LocalWeightedOrbital.splitOrbital_eq_zero_of_not_exists_norm_eq_of_areMatchingLocal7 below · cited by 1 · depth 38 - Split orbital integral at diag(a,b) in Iwasawa form
AutomorphicForm.LocalWeightedOrbital.eq_mul_splitOrbital_of_isOrbitalIntegral_diagUnits25 below · cited by 1 · depth 39
AutomorphicForm.PseudoEisensteinSlab 1
- L² bound for pseudo-Eisenstein series on a determinant slab
AutomorphicForm.PseudoEisensteinSlab.eLpNorm_pseudoEisenstein_le_of_adelicHeight_mem_Icc31 below · cited by 2 · depth 35
AutomorphicForm.RankinSelberg 18
- Rankin–Selberg package for one continuous GL₂ cusp realisation
AutomorphicForm.RankinSelberg.exists_testData_analyticOnNhd_sub_one_half_mul_peterssonIntegral_and_hasProd_rsEulerPoly_self552 below · cited by 2 · depth 18 - Euler factorisation of the unfolded Rankin–Selberg quotient integral
AutomorphicForm.RankinSelberg.exists_hasProd_quotientIntegral_eq_sPartIntegral_mul_of_shell_recursion32 below · cited by 2 · depth 19 - Rankin–Selberg test data: bad-place part analytic and positive past 1/2
AutomorphicForm.RankinSelberg.exists_testData_sPartIntegral_self_analyticOnNhd_re_pos390 below · cited by 1 · depth 19 - Holomorphy and positivity of the S-part Rankin–Selberg integral
AutomorphicForm.RankinSelberg.analyticOnNhd_sPartIntegral_and_pos_of_shell_surgery32 below · cited by 1 · depth 20 - Rankin–Selberg package for a pair of cusp realisations
AutomorphicForm.RankinSelberg.exists_testData_analyticOnNhd_sub_mul_peterssonIntegral_and_hasProd_rsEulerPoly_pair593 below · cited by 1 · depth 20 - Section law, shell majorant and base value after shell surgery
AutomorphicForm.RankinSelberg.exists_finset_norm_whittakerCoefficient_sq_mul_norm_section_le_shell_indicator_of_shell_surgery10 below · cited by 1 · depth 21 - Rankin–Selberg test data: bad part analytic, non-zero at centre
AutomorphicForm.RankinSelberg.exists_testData_sPartIntegral_pair_analyticOnNhd_ne_zero420 below · cited by 1 · depth 21 - Unfolded Rankin–Selberg S-part as a torus integral
AutomorphicForm.RankinSelberg.lintegral_sPart_quotientIntegrand_eq_mul_lintegral_torus_and_sPartIntegral_eq19 below · cited by 2 · depth 21 - S-part Rankin–Selberg integral: continuation past 1/2 and non-vanishing
AutomorphicForm.RankinSelberg.analyticOnNhd_sPartIntegral_pair_and_ne_zero_of_ball_surgery35 below · cited by 1 · depth 22 - Non-vanishing of an archimedean Rankin–Selberg torus pairing
AutomorphicForm.RankinSelberg.exists_archTranslate_isArchKFinite_equivariant_integral_mul_torusIntegral_whittakerCoefficient_ne_zero30 below · cited by 1 · depth 22 - Non-vanishing Rankin–Selberg torus pairing against a non-negative K-finite datum
AutomorphicForm.RankinSelberg.exists_archTranslate_isArchKFinite_equivariant_nonneg_integral_mul_torusIntegral_whittakerCoefficient_ne_zero_of_eq_one31 below · cited by 1 · depth 22 - Shell majorant for a surgered Whittaker–section integrand
AutomorphicForm.RankinSelberg.exists_finset_norm_whittakerCoefficient_sq_mul_norm_section_le_shell_indicator_of_shell_surgery_of_section_law7 below · cited by 1 · depth 22 - Induced sections on the torus: φₛ(diag(t,1)k)=‖t‖^{s+1/2}φₛ(k)
AutomorphicForm.RankinSelberg.section_diagOne_mul_eq_ideleNorm_cpow_mul_of_isInducedSection_etaFst_etaSnd0 below · cited by 1 · depth 22 - Shell surgery preserves the Whittaker coefficient at diag(t₀,1)k₀
AutomorphicForm.RankinSelberg.whittakerCoefficient_diagOne_mul_mul_inv_finEmbed_eq_of_shell_surgery0 below · cited by 1 · depth 22 - Analyticity of an archimedean torus Rankin–Selberg pairing
AutomorphicForm.RankinSelberg.analyticOnNhd_integral_archTorus_pair11 below · cited by 1 · depth 23 - Ball-surgered torus integral evaluated past the centre
AutomorphicForm.RankinSelberg.exists_integral_torus_pair_eq_mul_integral_archTorus_of_ball_surgery22 below · cited by 1 · depth 23 - Absolute convergence of the ball-surgered torus S-part integral
AutomorphicForm.RankinSelberg.lintegral_torus_pair_lt_top_of_ball_surgery15 below · cited by 2 · depth 23 - Pointwise torus evaluation of a ball-surgered Rankin–Selberg integrand
AutomorphicForm.RankinSelberg.whittakerCoefficient_mul_conj_mul_section_diagOne_mul_eq_of_ball_surgery7 below · cited by 2 · depth 24
AutomorphicForm.RealIwasawa 7
- Smoothness and uniform derivative bounds for the real Iwasawa rotation factor
AutomorphicForm.RealIwasawa.contDiff_and_exists_bound_iteratedFDeriv_kR_apply0 below · cited by 2 · depth 23 - Weight-k real Whittaker integral: continuity, holomorphy, rapid decay
AutomorphicForm.RealIwasawa.continuousOn_differentiableOn_norm_le_polyDecay_weightFourierIntegral2 below · cited by 7 · depth 23 - Rapid decay of the Fourier integral of r_g^{-u}P
AutomorphicForm.RealIwasawa.norm_fourierIntegral_cpow_rad_mul_le_polyDecay0 below · cited by 1 · depth 23 - Uniform polynomial decay of Fourier transforms of rad^{-u}P
AutomorphicForm.RealIwasawa.exists_forall_norm_fourierIntegral_cpow_rad_mul_le_polyDecay_of_isCompact0 below · cited by 1 · depth 24 - Entire continuation of the weight-k archimedean Whittaker integral
AutomorphicForm.RealIwasawa.exists_entire_weightFourierIntegral_norm_le_rpow_neg_mul_polyDecay3 below · cited by 1 · depth 30 - Entire continuation of the real weight–Fourier integral for t ≠ 0
AutomorphicForm.RealIwasawa.exists_entire_eq_weightFourierIntegral_norm_le_of_ne_zero3 below · cited by 2 · depth 32 - Uniform strip bound for the real-place weight–Fourier integral
AutomorphicForm.RealIwasawa.exists_forall_norm_weightFourierIntegral_continuation_le_mul_pow_abs_im_mul_pow_of_re_mem_Icc3 below · cited by 1 · depth 34
AutomorphicForm.SatakeCombination 7
- Slot combination of unipotent edge moments at one place
AutomorphicForm.SatakeCombination.sum_slotCoeff_mul_unipotentEdgeMoment_eq_mul_sum_laurentCoeff_edge0 below · cited by 1 · depth 29 - Slot combination of unipotent moments at one place
AutomorphicForm.SatakeCombination.sum_slotCoeff_mul_unipotentMoment_eq_mul_laurentCoeff_zero0 below · cited by 1 · depth 29 - Satake slot combination equals tilted Laurent symbol coefficient
AutomorphicForm.SatakeCombination.mul_sum_slotCoeff_div_pow_mul_ite_apply_T_add_T_inv_pow_eq_ite_sqrt_mul_pow_mul_zpow_neg0 below · cited by 1 · depth 32 - Satake word comparison at an inertia-degree-one place
AutomorphicForm.SatakeCombination.sum_slotCoeff_mul_sum_indicator_heckeWord_eq_sum_indicator_map_heckeWord_of_inertiaDeg_eq_one8 below · cited by 1 · depth 33 - Twisted shell sum equals ℓ times base-changed shell sum
AutomorphicForm.SatakeCombination.sum_twistedShell_heckeWord_eq_mul_sum_shell_baseChange_of_lt0 below · cited by 2 · depth 33 - Twisted shell sum equals ℓ times base-changed shell sum, even case
AutomorphicForm.SatakeCombination.sum_twistedShell_heckeWord_eq_mul_sum_shell_baseChange_of_two_mul_eq0 below · cited by 2 · depth 33 - Twisted shell value equals ℓ times the base-change shell value
AutomorphicForm.SatakeCombination.twistedShellValue_eq_mul_shellValue2 below · cited by 1 · depth 33
AutomorphicForm.SiegelCovering 3
- Adelic Siegel covering for GL₂ over a number field
AutomorphicForm.SiegelCovering.exists_finset_coversModCentre_iUnion_mul_centreCutSiegelSet3 below · cited by 41 · depth 11 - Production Siegel domain over ℚ covers modulo centre
AutomorphicForm.SiegelCovering.coversModCentre_productionPinsGeneral_D_rat2 below · cited by 31 · depth 13 - Centre-cut Siegel sets cover adelic GL₂ over ℚ
AutomorphicForm.SiegelCovering.centreCutSiegelSet_coversModCentre_rat1 below · cited by 6 · depth 14
AutomorphicForm.SmoothCusp 1
- No Hecke coset eigenfunction at level bot
AutomorphicForm.SmoothCusp.not_isHeckeCosetEigenfunctionAt_levelOne_bot_inf_finiteAdelicGL2Subgroup1 below · cited by 1 · depth 19
AutomorphicForm.SmoothCuspRealizationAt 26
- Central character: idele class character admitting the level as modulus
AutomorphicForm.SmoothCuspRealizationAt.isIdeleClassChar_and_admitsModulus_level_and_continuous_of_genuine0 below · cited by 20 · depth 15 - Central character determined by Hecke eigenvalues away from a finite set
AutomorphicForm.SmoothCuspRealizationAt.centralChar_eq_of_agreesAwayFromFinite4 below · cited by 16 · depth 16 - Unimodular central eigenvalues: central character has modulus the idelic norm
AutomorphicForm.SmoothCuspRealizationAt.norm_centralChar_eq_ideleNorm_of_forall_norm_b_eq_one9 below · cited by 5 · depth 16 - A bad set for a smoothed cusp realisation, with coset data
AutomorphicForm.SmoothCuspRealizationAt.exists_finset_badSet_rightConv_section31 below · cited by 1 · depth 17 - Polynomial bounds on Hecke eigenvalues from moderate growth
AutomorphicForm.SmoothCuspRealizationAt.exists_forall_norm_a_le_rpow_and_norm_b_le_rpow_of_moderateGrowth4 below · cited by 1 · depth 17 - Continuous cuspidal realisations are not translation eigenvectors at v
AutomorphicForm.SmoothCuspRealizationAt.not_exists_forall_apply_mul_heckeGen_eq_of_continuous2 below · cited by 1 · depth 17 - Hecke eigenvalue relation for Whittaker coefficients at a good place
AutomorphicForm.SmoothCuspRealizationAt.sum_whittakerCoefficient_mul_placeEmbed_repSome_add_eq_a_mul_whittakerCoefficient1 below · cited by 2 · depth 17 - Central eigenvalue bᵥ shifts the Whittaker coefficients
AutomorphicForm.SmoothCuspRealizationAt.whittakerCoefficient_mul_placeEmbed_scalarPi_eq_b_mul_whittakerCoefficient0 below · cited by 2 · depth 17 - Genericity at places off the level and exceptional set
AutomorphicForm.SmoothCuspRealizationAt.a_sq_ne_b_mul_of_not_dvd_level_of_not_mem_exceptionalSet25 below · cited by 2 · depth 18 - Paired Whittaker coefficients follow the Hecke recursion at good places
AutomorphicForm.SmoothCuspRealizationAt.whittakerCoefficient_heckeGen_pow_mul_conj_eq_heckeRecursionSeq_mul_of_rightConv_sum_translate_pair14 below · cited by 4 · depth 18 - Central exponent at a real place: positive scalars act by t^{c₀}
AutomorphicForm.SmoothCuspRealizationAt.exists_cpow_centralExponent_of_isReal2 below · cited by 2 · depth 19 - Admissible unitary untwist of a cuspidal central character
AutomorphicForm.SmoothCuspRealizationAt.exists_isAdmissibleTwist_eq_centralChar_mul_ideleNorm_inv11 below · cited by 3 · depth 19 - Rankin–Selberg Euler product for a cuspidal-constituent cusp realization
AutomorphicForm.SmoothCuspRealizationAt.exists_lt_one_meromorphicOn_analyticAt_hasProd_rsEulerPoly_self_of_isCuspConstituent553 below · cited by 1 · depth 19 - Partial Rankin–Selberg Euler product: meromorphy past s=1 and rigidity
AutomorphicForm.SmoothCuspRealizationAt.exists_lt_one_meromorphicOn_hasProd_rsEulerPoly_and_agreesAwayFromFinite_pair_of_isCuspConstituent595 below · cited by 1 · depth 19 - Archimedean smoothing of a cuspidal realisation at a real place
AutomorphicForm.SmoothCuspRealizationAt.exists_rightConv_ne_zero_mem_isotypicCuspSubmodule_mem_archCutSubmodule_hasArchCharacterAt_of_isReal77 below · cited by 5 · depth 19 - Twisting a cusp realization by ‖det‖_A^t
AutomorphicForm.SmoothCuspRealizationAt.exists_twist_rpow_absNorm_exceptionalSet_eq_toFun_eq_ideleNorm_det_rpow_mul10 below · cited by 6 · depth 19 - Non-vanishing first Whittaker coefficient at a torus point trivial outside S
AutomorphicForm.SmoothCuspRealizationAt.exists_mem_maximalCompactAt_whittakerCoefficient_rightConv_diagOne_mul_ne_zero43 below · cited by 2 · depth 20 - Non-vanishing first Whittaker coefficient over ℚ
AutomorphicForm.SmoothCuspRealizationAt.exists_whittakerCoefficient_one_ne_zero_of_continuous_foldr_archDerivAt_rat23 below · cited by 2 · depth 20 - Continuous cusp realisations admit no Hecke-generator translation eigenvalue
AutomorphicForm.SmoothCuspRealizationAt.not_exists_forall_apply_mul_heckeGen_eq_of_isGenuineCuspRealizationAt2 below · cited by 2 · depth 20 - Unramified package at a good place for smoothed translate sums
AutomorphicForm.SmoothCuspRealizationAt.unramified_package_rightConv_sum_translate12 below · cited by 4 · depth 20 - Export package for translates of a smoothed cuspidal realisation
AutomorphicForm.SmoothCuspRealizationAt.exports_rightConv_sum_translate_of_isCuspConstituent109 below · cited by 1 · depth 22 - Modulus of the central character is ‖·‖^σ
AutomorphicForm.SmoothCuspRealizationAt.norm_centralChar_eq_ideleNorm_rpow_of_forall_norm_b_eq9 below · cited by 2 · depth 22 - Equal central characters from eigensystems agreeing almost everywhere
AutomorphicForm.SmoothCuspRealizationAt.centralChar_eq_of_agreesAwayFromFinite_principal4 below · cited by 1 · depth 28 - Central character of a principal-level smooth cusp realization
AutomorphicForm.SmoothCuspRealizationAt.isIdeleClassChar_and_admitsModulus_level_and_continuous_of_genuine_principal0 below · cited by 2 · depth 28 - Hecke recursion for Whittaker coefficients at a good place
AutomorphicForm.SmoothCuspRealizationAt.sum_whittakerCoefficient_mul_placeEmbed_repSome_add_eq_a_mul_whittakerCoefficient_principal1 below · cited by 1 · depth 28 - Central uniformizer scalar at a good place scales Whittaker coefficients
AutomorphicForm.SmoothCuspRealizationAt.whittakerCoefficient_mul_placeEmbed_scalarPi_eq_b_mul_whittakerCoefficient_principal0 below · cited by 1 · depth 28
AutomorphicForm.SplitPlace 4
- Split coordinates on GL₂(L⊗_K A): shift, norm string, Haar
AutomorphicForm.SplitPlace.continuous_coords_and_coords_sigmaGL_and_coords_normString_and_exists_pos_map_coords_eq_smul_pi0 below · cited by 8 · depth 30 - Split coordinates of a matrix placed at a split place
AutomorphicForm.SplitPlace.exists_ringEquiv_coords_semiLocalComponent_localEmbed_eq_mulSingle0 below · cited by 1 · depth 33 - Split-place coordinates: integrality, weights and Haar normalisation
AutomorphicForm.SplitPlace.mem_semiLocalIntegralSet_iff_coords_and_semiLocalWeight_eq_sum_and_map_coords_semiLocalHaar2 below · cited by 1 · depth 33 - Split-place coordinates agree with the completions L_w
AutomorphicForm.SplitPlace.exists_equiv_extension_algEquiv_forall_psi_eq_and_mem_adicCompletionIntegers_iff_and_norm_eq0 below · cited by 1 · depth 34
AutomorphicForm.StandardKernel 1
- Haar measure on (ℚ⊗ℝ)^× pushes forward to κ |y|⁻¹dy
AutomorphicForm.StandardKernel.exists_pos_map_realCoord_eq_smul_volume_withDensity_abs_inv0 below · cited by 10 · depth 21
AutomorphicForm.TransversalMeasure 6
- Measurable fundamental domain for the principal K-ideles in A_L^×
AutomorphicForm.TransversalMeasure.exists_measurableSet_isFundamentalDomain_idelesBaseChange_principal10 below · cited by 1 · depth 30 - Saturation criterion at an unramified place, cyclic case
AutomorphicForm.TransversalMeasure.mem_saturatedUnits_of_forall_ne_valued_semiLocalUnitComponent_congr_mul_inv_eq_one_unram3 below · cited by 1 · depth 31 - Transversal measure identity over a K^×-fundamental domain
AutomorphicForm.TransversalMeasure.setLIntegral_fundamentalDomain_inter_saturated_eq_mul_setLIntegral_lintegral_sum_of_transversal0 below · cited by 1 · depth 32 - Galois action on archimedean semi-local idele components
AutomorphicForm.TransversalMeasure.archSemiLocalIdele_unitsAct_eq_placeEquivAlg_congr_symm1 below · cited by 2 · depth 33 - Collecting the archimedean factors of a factorising idele measure
AutomorphicForm.TransversalMeasure.exists_forall_lintegral_and_integral_eq_mul_prod_of_forall_prod_archSemiLocalIdele26 below · cited by 1 · depth 33 - Assembling the archimedean factors of a transversal measure
AutomorphicForm.TransversalMeasure.exists_forall_lintegral_eq_lintegral_mul_prod_of_forall_prod_archSemiLocalIdele24 below · cited by 1 · depth 34
AutomorphicForm.TwistedBruhat 30
- Vanishing of the twisted unipotent term off the saturated set
AutomorphicForm.TwistedBruhat.apply_unipotent_diagOne_act_eq_zero_of_not_mem_saturated_of_isSemiLocalFactorization_unram8 below · cited by 1 · depth 30 - Twisted unipotent term: transversal descent to rank-one Tate data
AutomorphicForm.TwistedBruhat.exists_forall_integral_transversal_finsum_tracePushforward_sub_eq_finsum_indicator_prod_twistedLocalFactor_sub_unram79 below · cited by 1 · depth 30 - Transversal descent and dilation of the unfolded unipotent term
AutomorphicForm.TwistedBruhat.integrableOn_and_integral_finsum_tracePushforward_sub_eq_sum_mul_setIntegral_rankOne_of_transversal16 below · cited by 1 · depth 30 - Centre removal in the Iwasawa integral of the twisted cusp kernel
AutomorphicForm.TwistedBruhat.integral_iwasawa_indicator_cuspKernel_sub_cuspTruncation_eq_measure_mul_integral_of_sigmaInvariant_ed221 below · cited by 1 · depth 30 - Removing the central variable from the unipotent-type Iwasawa lower integral
AutomorphicForm.TwistedBruhat.lintegral_iwasawa_indicator_tsum_tsum_enorm_sub_eq_measure_mul_lintegral_of_sigmaInvariant20 below · cited by 1 · depth 30 - Unfolding the unipotent term along centre, torus and trace
AutomorphicForm.TwistedBruhat.lintegral_ne_top_and_integral_iwasawa_cuspKernel_sub_cuspTruncation_eq_mul_integral_finsum_tracePushforward_sub118 below · cited by 1 · depth 30 - Transversal integral of the unramified twisted unipotent term as a pure tensor
AutomorphicForm.TwistedBruhat.exists_forall_integral_transversal_tracePushforward_eq_indicator_prod_twistedLocalFactor_unram76 below · cited by 1 · depth 31 - Lattice sum and constant term commute with transversal integrals
AutomorphicForm.TwistedBruhat.forall_integral_transversal_finsum_tracePushforward_sub_eq_finsum_integral_transversal_sub_unram42 below · cited by 1 · depth 31 - Transversal descent of the unipotent fold to rank-one integrals
AutomorphicForm.TwistedBruhat.integrableOn_and_integral_unipotentFold_eq_sum_mul_setIntegral_rankOne_of_invariance_of_dilation_of_ne_top2 below · cited by 1 · depth 31 - Fibrewise collapse of the twisted cusp kernel Iwasawa integral
AutomorphicForm.TwistedBruhat.integral_iwasawa_cuspKernel_sub_cuspTruncation_eq_integral_tsum_normOneFibre_of_fibrewise5 below · cited by 1 · depth 31 - Unfolding norm-one fibres onto the unit fibre
AutomorphicForm.TwistedBruhat.integral_iwasawa_tsum_normOneFibre_eq_integral_unitFibre_of_fibrewise13 below · cited by 1 · depth 31 - Unfolding the twisted unipotent kernel along the trace fibration
AutomorphicForm.TwistedBruhat.lintegral_ne_top_and_integral_iwasawa_unitFibre_eq_mul_integral_finsum_tracePushforward_sub112 below · cited by 1 · depth 31 - Measurability of the twisted unipotent fold
AutomorphicForm.TwistedBruhat.measurable_unipotentFold4 below · cited by 1 · depth 31 - Central invariance of the ξ-folded truncated cusp kernel
AutomorphicForm.TwistedBruhat.setIntegral_mul_cuspKernel_sub_cuspTruncation_centralScalar_mul_eq_of_sigmaInvariant0 below · cited by 1 · depth 31 - Base-changed ideles fold out of the twisted Bruhat integral
AutomorphicForm.TwistedBruhat.unipotentFold_mul_idelesBaseChange_eq_mul_integral_finsum_tracePushforward_sub4 below · cited by 1 · depth 31 - Invariance of the twisted Bruhat fold under K^×
AutomorphicForm.TwistedBruhat.unipotentFold_mul_idelesBaseChange_map_algebraMap_eq8 below · cited by 1 · depth 31 - Word-independent factorisation of unramified unipotent twisted transversal integrals
AutomorphicForm.TwistedBruhat.exists_forall_integral_transversal_eq_indicator_mul_prod_unipotentOrbitalFn_unram69 below · cited by 1 · depth 32 - Effective support, uniform bound and continuity of the twisted unipotent integrand
AutomorphicForm.TwistedBruhat.exists_isCompact_forall_unipotentTwist_traceFibre_bound_and_eq_zero_unram39 below · cited by 3 · depth 32 - Torus and unipotent equivariance of twisted Borel fibre sums
AutomorphicForm.TwistedBruhat.finsum_fibre_eq_unitFibre_diagOne_inv_mul_and_unitFibre_unipotent_mul_eq5 below · cited by 1 · depth 32 - Unit-diagonal Bruhat fibres as sums over L
AutomorphicForm.TwistedBruhat.finsum_unitFibre_iwasawa_eq_finsum_trace_ne_zero_and_finsum_unitFibre_unipotent_eq_finsum3 below · cited by 1 · depth 32 - Unipotent merge: fundamental-domain integral as trace push-forward sum
AutomorphicForm.TwistedBruhat.integrableOn_and_setIntegral_finsum_trace_ne_zero_unipotentMerge_eq_mul_finsum_tracePushforward6 below · cited by 1 · depth 32 - Truncated twisted constant term integrated over a fundamental domain
AutomorphicForm.TwistedBruhat.integrableOn_and_setIntegral_indicator_constantTerm_unitFibre_eq_mul_ite_integral_tracePushforward10 below · cited by 1 · depth 32 - Fubini interchange of trace push-forward with transversal integrals
AutomorphicForm.TwistedBruhat.integral_transversal_tracePushforward_eq_tracePushforward_integral_of_bound2 below · cited by 1 · depth 32 - Almost every idele lies in the structured box
AutomorphicForm.TwistedBruhat.ae_mem_structuredBox_of_transversal0 below · cited by 2 · depth 33 - Smoothness and compact support of the archimedean unipotent integral
AutomorphicForm.TwistedBruhat.continuous_and_hasCompactSupport_and_contDiff_integral_archWord1 below · cited by 1 · depth 33 - Bounded Galois ratio confines transversal ideles to a compact set
AutomorphicForm.TwistedBruhat.exists_isCompact_forall_ae_mem_of_unitsAct_mul_inv_mem_of_transversal_unram37 below · cited by 1 · depth 33 - Compact confinement of the central variable in a twisted word
AutomorphicForm.TwistedBruhat.exists_isCompact_forall_archWord_eq_zero_of_not_mem0 below · cited by 1 · depth 33 - Joint archimedean confinement of norm-one ideles with bounded twisted ratio
AutomorphicForm.TwistedBruhat.exists_isCompact_forall_mem_of_forall_archFibre_mem_archNormOneUnits_of_map_mul_inv_mem5 below · cited by 1 · depth 33 - Compactness of twisted ratios on a norm shell
AutomorphicForm.TwistedBruhat.exists_isCompact_forall_mem_of_mem_smul_normOneUnits_of_congr_mul_inv_mem6 below · cited by 2 · depth 33 - Compactness of archimedean norm-one units with bounded σ-ratio
AutomorphicForm.TwistedBruhat.exists_isCompact_forall_mem_of_mem_archNormOneUnits_of_placeEquivAlg_congr_mul_inv_mem3 below · cited by 2 · depth 34
AutomorphicForm.WeylIntegrable 6
- Modulus of the big-component idele: archimedean height times index
AutomorphicForm.WeylIntegrable.Dy_eq_prod_mul_relIndex3 below · cited by 2 · depth 31 - Positivity of the adelic modulus D_y
AutomorphicForm.WeylIntegrable.Dy_pos0 below · cited by 2 · depth 31 - Dilated integral lattice depends only on the finite component
AutomorphicForm.WeylIntegrable.dilate_finPart_of_snd_eq0 below · cited by 1 · depth 31 - Uniform bound for induced sections on the big Bruhat cell
AutomorphicForm.WeylIntegrable.norm_apply_weyl_unipotent_le_uniform0 below · cited by 1 · depth 31 - Translation bound for negative powers of D_y
AutomorphicForm.WeylIntegrable.rpow_Dy_le_translate_of_le5 below · cited by 1 · depth 31 - Product of local norms equals index of integral dilate
AutomorphicForm.WeylIntegrable.finprod_norm_eq_relIndex_dilate2 below · cited by 1 · depth 32
AutomorphicForm.WhittakerModel 21
- Kirillov-model majorant for Whittaker functions on the torus
AutomorphicForm.WhittakerModel.exists_norm_diagOne_mul_le_of_irreducible_admissible2 below · cited by 6 · depth 19 - Local Whittaker vectors at p inherit the central character
AutomorphicForm.WhittakerModel.forall_mem_localSpaceAt_scalar_mul_eq_localChar_mul0 below · cited by 4 · depth 19 - Deep twist functional equation for GL₂ Whittaker torus integrals
AutomorphicForm.WhittakerModel.exists_torusZeta_dual_eq_stdRootNumberAt_mul_stdRootNumberAt_mul_of_admissible_of_le_of_norm_eq_one29 below · cited by 1 · depth 21 - Shell form of the local functional equation for deep twists
AutomorphicForm.WhittakerModel.exists_torusShell_eq_zero_and_torusShell_dual_eq_stdRootNumberAt_mul_of_mem_span25 below · cited by 2 · depth 22 - Shell vanishing and recurrence for admissible Whittaker spaces
AutomorphicForm.WhittakerModel.exists_polynomial_forall_diagZ_mul_eq_zero_and_sum_coeff_mul_eq_zero_of_admissible1 below · cited by 4 · depth 23 - Torus-shell vanishing and shell functional equation, deep twist
AutomorphicForm.WhittakerModel.exists_torusShell_eq_zero_and_torusShell_dual_eq_stdRootNumberAt_mul_of_mem_localLevelOne_top24 below · cited by 1 · depth 23 - Properties of the cyclic span of a local Whittaker function
AutomorphicForm.WhittakerModel.span_translates_stable_and_law_and_smooth_and_irreducible_and_central1 below · cited by 1 · depth 23 - Gauge bound for an admissible local Whittaker function on the torus
AutomorphicForm.WhittakerModel.exists_norm_diagUnits2_mul_le_and_eq_zero_of_admissible_of_centralChar4 below · cited by 8 · depth 26 - Translates of a Whittaker vector: smoothness, growth, shell recurrence
AutomorphicForm.WhittakerModel.forall_mem_span_smooth_and_law_and_central_and_growth_and_shellRecurrence4 below · cited by 4 · depth 26 - Unit shell at the Weyl element for a deep twist
AutomorphicForm.WhittakerModel.setIntegral_unitShell_diagOne_weyl_eq_stdRootNumberAt_mul_setIntegral_shell_of_admissible_of_le_of_norm_eq_one26 below · cited by 1 · depth 26 - Gauge bound for torus values of admissible Whittaker functions
AutomorphicForm.WhittakerModel.exists_forall_diagZ_mul_eq_zero_and_norm_le_mul_zpow_of_admissible3 below · cited by 6 · depth 27 - Kirillov model contains the compactly supported locally constant functions
AutomorphicForm.WhittakerModel.exists_mem_span_forall_diagOne_eq_of_shell_window_of_irreducible2 below · cited by 3 · depth 28 - Shell-window functions in the Whittaker translate span, ideal level
AutomorphicForm.WhittakerModel.exists_mem_span_forall_diagOne_eq_of_shell_window_of_localLevelOne3 below · cited by 8 · depth 28 - Cuspidal Whittaker space equals V(N)
AutomorphicForm.WhittakerModel.forall_mem_span_sub_unipotent_of_forall_diagOne_eq_zero_of_irreducible_of_admissible18 below · cited by 2 · depth 28 - Propagating a Whittaker gauge from the level-one subgroup to GL₂
AutomorphicForm.WhittakerModel.norm_diagUnits2_mul_le_of_forall_mem_localLevelOne_norm_diagUnits2_mul_le0 below · cited by 2 · depth 28 - Span of right translates of a local Whittaker function
AutomorphicForm.WhittakerModel.span_translates_stable_and_law_and_smooth_and_central1 below · cited by 2 · depth 28 - Kirillov injectivity for a Whittaker span at level K₁(N)
AutomorphicForm.WhittakerModel.eq_zero_of_forall_apply_diagOne_eq_zero_of_mem_span_of_localLevelOne11 below · cited by 8 · depth 29 - Whittaker functions on GL₂(ℚₚ) vanish for large |y|
AutomorphicForm.WhittakerModel.exists_forall_diagOne_eq_zero_of_lt_modulus5 below · cited by 3 · depth 29 - Right-translation stability of the local Whittaker space at p
AutomorphicForm.WhittakerModel.localSpaceAt_comp_mul_right_mem0 below · cited by 1 · depth 29 - Multiplicity one for torus-eigenfunctionals on a Whittaker space
AutomorphicForm.WhittakerModel.apply_mul_apply_eq_apply_mul_apply_of_forall_diagOne_eq_smul_of_transcendental15 below · cited by 1 · depth 31 - Finitely many vectors span V modulo its unipotent coinvariants
AutomorphicForm.WhittakerModel.exists_finset_span_mod_unipotentCoinvariants_of_irreducible_admissible0 below · cited by 1 · depth 32
AutomorphicForm.WindingDatum 5
- Winding data: bounded coefficients and an atom-free interpolating functional
AutomorphicForm.WindingDatum.exists_clm_noAtomicMass_forall_apply_fourier_eq_coeff8 below · cited by 1 · depth 27 - Realising finite smooth lattice sums as winding-datum coefficients
AutomorphicForm.WindingDatum.exists_forall_coeff_eq_sum_tsum_ite_of_contDiff_of_periodic5 below · cited by 1 · depth 31 - Uniform absolute bound for fibre coefficients of a winding datum
AutomorphicForm.WindingDatum.exists_forall_summable_norm_fibreTerm_and_norm_fibreCoeff_le1 below · cited by 4 · depth 32 - Pairing a finitely supported array against winding-datum coefficients
AutomorphicForm.WindingDatum.sum_mul_coeff_eq_tsum_mul_tsum2 below · cited by 2 · depth 32 - Kink-window lattice sums realised as winding-datum coefficients
AutomorphicForm.WindingDatum.exists_forall_coeff_eq_tsum_mul_tsum_ite_kinkWindow_of_contDiff_of_periodic_of_summable23 below · cited by 1 · depth 33
AutomorphicForm.WindowedSiegel 6
- Compactness of the centre-cut Siegel set under height caps
AutomorphicForm.WindowedSiegel.isCompact_centreCutSiegelSet_inter_heightCap0 below · cited by 10 · depth 15 - Siegel support property high in the cusp, with central twist
AutomorphicForm.WindowedSiegel.exists_forall_apply_one_zero_eq_zero_of_inv_mul_globalPoints_mul_mul_centralScalar_mem0 below · cited by 4 · depth 17 - Compact cover of the low part of a windowed Siegel set
AutomorphicForm.WindowedSiegel.exists_isCompact_cover_of_archHeight_le3 below · cited by 8 · depth 17 - Support lemma: high in the cusp forces γ₂₁=0
AutomorphicForm.WindowedSiegel.exists_forall_apply_one_zero_eq_zero_of_inv_mul_globalPoints_mul_mem0 below · cited by 4 · depth 24 - Height bound for non-triangular rational translates on Siegel translates
AutomorphicForm.WindowedSiegel.exists_forall_adelicHeight_globalPoints_mul_le_of_subset_iUnion_mul_centreCutSiegelSet1 below · cited by 7 · depth 26 - Weight bridge: sum_w m_wlog(topcdotrow/‖det‖²) versus archimedean heights
AutomorphicForm.WindowedSiegel.sum_mult_mul_log_topNormSq_mul_rowNormSq_div_eq_neg_log_archHeight_sub_log_archHeight_weyl_mul0 below · cited by 3 · depth 34